Quantum Dots

By Timothy Paik

Marcus DahlstromMichael Nip

Implementing Quantum Computers

bull Many implementations for quantum computing

bull Why solid statendash Scalabilityndash Decoherence is less of a problem

What is a quantum dot

bull In two words a semiconductor nanocrystal

bull Easily tunable by changing the size and composition of the nanocrystal

Gallium Arsenide Quantum Dots

bull Gallium arsenide is a III-V semiconductorndash Higher saturated electron velocity and higher

electron mobility than siliconndash Gallium arsenide can emit and absorb light

unlike siliconbull No silicon laser is possible (or has been made yet)

Energy Band Levels

bull Electrons exist in discrete energy levels in bulk semiconductor materialndash There exists a forbidden

range of energy levels in any material called the band gap

Energy Band Levels

bull By absorbing some sort of stimulus (in light or heat form) an electron can rise to the conduction band from the valence bandndash This action leaves behind a

ldquoholerdquo in the valence band The hole and the electron together are called an exciton

Energy Band Levels

bull The average distance between an electron and a hole in a exciton is called the Excited Bohr Radius

bull When the size of the semiconductor falls below the Bohr Radius the semiconductor is called a quantum dot

Tuning Quantum Dots

bull By changing size shape and composition quantum dots can change their absorptive and emissive properties dramatically

Manufacturing methods

bull Electron beam lithography

bull Molecular beam epitaxy

Electron Beam Lithography

bull Electrons are accelerated out of an electron gun and sent through condenser lens optics directly onto a wafer

bull λ = (123 Aring radicV)bull Advantages

ndash generation of micron and submicron resist geometries

ndash greater depth of focus than optical lithography

ndash masks are unnecessaryndash Optical diffraction limit is not a

real concern

Electron Beam Lithography

bull Disadvantage(s)ndash The lithography is serial

(masks arenrsquot used instead the beam itself sweeps across the wafer) =gt Comparatively low throughput ~5 wafers per hour at less than 1 micrometer resolution

ndash The proximity effect Electrons scatter because they are relatively low in mass reducing the resolution

bull Heavy ion lithography has been proposed but still is in development stages

Molecular Beam Epitaxy

bull Molecular beam epitaxy (MBE) is the deposition of one or more pure materials onto a single crystal wafer one layer of atoms at a time in order to form a perfect crystalndash This is done by evaporating each of the elements to

combine then condensing them on top of the waferndash The word ldquobeamrdquo means that the evaporated atoms

only meet each other on the wafer

Spin Quantum Computing

Qubit information is stored in the spin state of an electron in an artificial atom

AdvantagesLong decoherence time

Future Scalabilty

Artifical atoms are bigger than regular atoms therefore easier to manipulate

Decoherence time ~ 100ns

bull Time before the quantum mechanical system starts acting in a classical way with its complex environment

bull The state of the system has not yet collapsed due to (unwanted) environmental effects

bull Spin - DT are 100 as long as for the Excitonbull Need to SWITCH 104 during DT for reliable error

correction This requirement is met

Artificial Atombull Double Barrier

Heterostructurebull Dot In005Ga095Asbull Source ampDrain GaAsbull 2D Electron Gasbull Confine with gate biasbull D ~ Fermi wavelength

rarr Discrete energy levels

Adding Electrons changing Vgate

bull 2D-Harmonic Oscillator

bull Shell structure as in atoms

bull Magic Numbers 2 6 12

bull To add ldquoevenrdquo electron requires only additional Coulomb energy

Comparison with Hydrogenbull Artificial Atom

Energy levels ~ 1meV

Size ~ 10μm

Weak magnetic fields can affect energy levels

bull Hydrogen

Energy levels ~ 1eV

Size ~ 1Aring

Only strong magnetic fields can perturb energy levels

Factor 1000

Tuning the Quantum Dot

bull Tune so we have one valence electron

bull Initial state can be set by applying homogeneous magnetic field rarr |0gt

bull Low temperature kT lt ΔE (state gap)

bull Now we have defined our single qubit

Energy

position

Gate bias

Spin up - electron

Unoccupied state

Single Qubit manipulation

bull Unitary operations can be made by applying a local magnetic field H

ZE = -μB = g

μB SB

bull MF microscopebull AF microscopebull Sub grid of currentbull Magnetic dotsbull Etc

(Magnetic force microscope tip)

Two Qubit Manipulation

bull Complete set of logic requires a CNOT

bull Dots are placed so close that they overlap and interact

bull Hspin

= J(t)S1S

2

Exchange couplingJ(tEB) = E

triplet -E

singlet

(4th order Harmonic Oscillator)

Ground State Splitting (J = Et ndash E

s)

bull 2 coupled fermions must have an total anti-symmetric wave function

bull Lowest coupled state is the singlet It has a symmetric spatial wave function and an anti symmetric spin (Coulomb dominates)|ψ

sgt ~ (|12gt + |21gt) (|darruarrgt - |uarrdarrgt)

bull The triplet states are (|darrdarrgt)|ψ

tgt ~ (|12gt - |21gt) (|darruarrgt + |uarrdarrgt)

(|uarruarrgt)bull lt1|2gt ne 0 |igt is spatial wf Coulomb dominates

Solving J(B(t)) Exchange Coupling

bull Different solutions Heitler-London Hund-Mulliken Hubbard

bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations

SWAP - gatebull Assume J can be pulsed

J(t) = 0 J0

Formula 1

Formula 2

bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]

XOR ~ CNOT

bull Formula 3

bull Requirements Spin rotations about the z-axis Squareroot of U

swap

Read out Memory

bull Assume dot with an electron with some information stored in spin-state

bull Connect two leads to dot

bull Apply a small bias (ΔV) rarr Current (i)Energy

position

Gate bias

Spin up - electron

Unoccupied state

i

Another Spin up electron enters dot

bull Pauli principle forces electrons with spin up to occupy the higher energy state

bull Negligible chance of tunneling

E

position

Gate bias

Spin up - electron

Higher energy level(forbidden classically)

i=0

Spin down electron enters dot

bull Pauli principle allows the new electron to join the same energy level as the original electron

bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow

E

position

Gate bias

Spin up - electron

Unoccupied state

ine0

Read out Memory

bull We have a way of measuring the spin state of an electron in a quantum dot

bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties

bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique

bull The electron current can be specialized (we can aim its spin to make measurement efficient)

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

The Physical System Excitons Trapped in GaAs Quantum Dots

bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid

bull Often formed when photons excite electrons from the valence band into the conduction band

bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses

bull Decay by radiating photons Decay time ~50ps-1ns

bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt

Initialization

bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to

prevent thermal excitations

bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the

register

Relatively Long Decoherence Times

bull Mechanismsndash Radiative Decay ~10ps-1ns

bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations

bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral

bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps

or less

A Universal set of Quantum Gates

bull Single Qubit Rotations through laser induced Rabi Oscillations

bull CNOT operations through dipole interactions and laser excitation

Single Qubit Gates Rabi Flopping

bull Light-particle interaction is characterized by the product of the dipole moment and the electric field

μbullE(t)= ħR(t)

Where R(t) is the Rabi frequency and the pulse area is given by

Θ(t)=intR(t)dt

and the state at time t is then given by

Cos(Θ2)|0gt+Sin(Θ2)|1gt

Stufler et al

Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses

Cos(Θ2)|0gt+Sin(Θ2)|1gt

|1gt =gt electric charge

=gtPhotocurrent (PC)PC~Sin2(Θ2)

π-pulse corresponds to a population inversion

CNOT Dipole Coupling

Nearest neighbor interactions alter the energy states

Effective energy Ersquoi = Ei + sumjnei ∆Eij nj

Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied

Overcoming Short Interaction Distances

bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs

bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots

bull Optical Cavity coupling and fiber optical interconnects have also been proposed

Read Out of Specified Qubit States

bull Optical readoutExcitons decay spontaneously and the resulting radiation can be

measured

Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD

Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination

In either case repeated measurements have to be made A single shot readout still needs to be developed

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

• Quantum Dots
• Implementing Quantum Computers
• What is a quantum dot
• Gallium Arsenide Quantum Dots
• Energy Band Levels
• Slide 6
• Slide 7
• Tuning Quantum Dots
• Manufacturing methods
• Electron Beam Lithography
• Slide 11
• Molecular Beam Epitaxy
• Spin Quantum Computing
• Decoherence time ~ 100ns
• Artificial Atom
• Adding Electrons changing Vgate
• Comparison with Hydrogen
• Tuning the Quantum Dot
• Single Qubit manipulation
• Two Qubit Manipulation
• Ground State Splitting (J = Et ndash Es)
• Solving J(B(t)) Exchange Coupling
• SWAP - gate
• XOR ~ CNOT
• Read out Memory
• Another Spin up electron enters dot
• Spin down electron enters dot
• Slide 28
• 5 DiVincenzo QC Criteria
• The Physical System Excitons Trapped in GaAs Quantum Dots
• Initialization
• Relatively Long Decoherence Times
• A Universal set of Quantum Gates
• Single Qubit Gates Rabi Flopping
• Stufler et al
• CNOT Dipole Coupling
• Overcoming Short Interaction Distances
• Read Out of Specified Qubit States
• Slide 39

Implementing Quantum Computers

bull Many implementations for quantum computing

bull Why solid statendash Scalabilityndash Decoherence is less of a problem

What is a quantum dot

bull In two words a semiconductor nanocrystal

bull Easily tunable by changing the size and composition of the nanocrystal

Gallium Arsenide Quantum Dots

bull Gallium arsenide is a III-V semiconductorndash Higher saturated electron velocity and higher

electron mobility than siliconndash Gallium arsenide can emit and absorb light

unlike siliconbull No silicon laser is possible (or has been made yet)

Energy Band Levels

bull Electrons exist in discrete energy levels in bulk semiconductor materialndash There exists a forbidden

range of energy levels in any material called the band gap

Energy Band Levels

bull By absorbing some sort of stimulus (in light or heat form) an electron can rise to the conduction band from the valence bandndash This action leaves behind a

ldquoholerdquo in the valence band The hole and the electron together are called an exciton

Energy Band Levels

bull The average distance between an electron and a hole in a exciton is called the Excited Bohr Radius

bull When the size of the semiconductor falls below the Bohr Radius the semiconductor is called a quantum dot

Tuning Quantum Dots

bull By changing size shape and composition quantum dots can change their absorptive and emissive properties dramatically

Manufacturing methods

bull Electron beam lithography

bull Molecular beam epitaxy

Electron Beam Lithography

bull Electrons are accelerated out of an electron gun and sent through condenser lens optics directly onto a wafer

bull λ = (123 Aring radicV)bull Advantages

ndash generation of micron and submicron resist geometries

ndash greater depth of focus than optical lithography

ndash masks are unnecessaryndash Optical diffraction limit is not a

real concern

Electron Beam Lithography

bull Disadvantage(s)ndash The lithography is serial

(masks arenrsquot used instead the beam itself sweeps across the wafer) =gt Comparatively low throughput ~5 wafers per hour at less than 1 micrometer resolution

ndash The proximity effect Electrons scatter because they are relatively low in mass reducing the resolution

bull Heavy ion lithography has been proposed but still is in development stages

Molecular Beam Epitaxy

bull Molecular beam epitaxy (MBE) is the deposition of one or more pure materials onto a single crystal wafer one layer of atoms at a time in order to form a perfect crystalndash This is done by evaporating each of the elements to

combine then condensing them on top of the waferndash The word ldquobeamrdquo means that the evaporated atoms

only meet each other on the wafer

Spin Quantum Computing

Qubit information is stored in the spin state of an electron in an artificial atom

AdvantagesLong decoherence time

Future Scalabilty

Artifical atoms are bigger than regular atoms therefore easier to manipulate

Decoherence time ~ 100ns

bull Time before the quantum mechanical system starts acting in a classical way with its complex environment

bull The state of the system has not yet collapsed due to (unwanted) environmental effects

bull Spin - DT are 100 as long as for the Excitonbull Need to SWITCH 104 during DT for reliable error

correction This requirement is met

Artificial Atombull Double Barrier

Heterostructurebull Dot In005Ga095Asbull Source ampDrain GaAsbull 2D Electron Gasbull Confine with gate biasbull D ~ Fermi wavelength

rarr Discrete energy levels

Adding Electrons changing Vgate

bull 2D-Harmonic Oscillator

bull Shell structure as in atoms

bull Magic Numbers 2 6 12

bull To add ldquoevenrdquo electron requires only additional Coulomb energy

Comparison with Hydrogenbull Artificial Atom

Energy levels ~ 1meV

Size ~ 10μm

Weak magnetic fields can affect energy levels

bull Hydrogen

Energy levels ~ 1eV

Size ~ 1Aring

Only strong magnetic fields can perturb energy levels

Factor 1000

Tuning the Quantum Dot

bull Tune so we have one valence electron

bull Initial state can be set by applying homogeneous magnetic field rarr |0gt

bull Low temperature kT lt ΔE (state gap)

bull Now we have defined our single qubit

Energy

position

Gate bias

Spin up - electron

Unoccupied state

Single Qubit manipulation

bull Unitary operations can be made by applying a local magnetic field H

ZE = -μB = g

μB SB

bull MF microscopebull AF microscopebull Sub grid of currentbull Magnetic dotsbull Etc

(Magnetic force microscope tip)

Two Qubit Manipulation

bull Complete set of logic requires a CNOT

bull Dots are placed so close that they overlap and interact

bull Hspin

= J(t)S1S

2

Exchange couplingJ(tEB) = E

triplet -E

singlet

(4th order Harmonic Oscillator)

Ground State Splitting (J = Et ndash E

s)

bull 2 coupled fermions must have an total anti-symmetric wave function

bull Lowest coupled state is the singlet It has a symmetric spatial wave function and an anti symmetric spin (Coulomb dominates)|ψ

sgt ~ (|12gt + |21gt) (|darruarrgt - |uarrdarrgt)

bull The triplet states are (|darrdarrgt)|ψ

tgt ~ (|12gt - |21gt) (|darruarrgt + |uarrdarrgt)

(|uarruarrgt)bull lt1|2gt ne 0 |igt is spatial wf Coulomb dominates

Solving J(B(t)) Exchange Coupling

bull Different solutions Heitler-London Hund-Mulliken Hubbard

bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations

SWAP - gatebull Assume J can be pulsed

J(t) = 0 J0

Formula 1

Formula 2

bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]

XOR ~ CNOT

bull Formula 3

bull Requirements Spin rotations about the z-axis Squareroot of U

swap

Read out Memory

bull Assume dot with an electron with some information stored in spin-state

bull Connect two leads to dot

bull Apply a small bias (ΔV) rarr Current (i)Energy

position

Gate bias

Spin up - electron

Unoccupied state

i

Another Spin up electron enters dot

bull Pauli principle forces electrons with spin up to occupy the higher energy state

bull Negligible chance of tunneling

E

position

Gate bias

Spin up - electron

Higher energy level(forbidden classically)

i=0

Spin down electron enters dot

bull Pauli principle allows the new electron to join the same energy level as the original electron

bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow

E

position

Gate bias

Spin up - electron

Unoccupied state

ine0

Read out Memory

bull We have a way of measuring the spin state of an electron in a quantum dot

bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties

bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique

bull The electron current can be specialized (we can aim its spin to make measurement efficient)

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

The Physical System Excitons Trapped in GaAs Quantum Dots

bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid

bull Often formed when photons excite electrons from the valence band into the conduction band

bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses

bull Decay by radiating photons Decay time ~50ps-1ns

bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt

Initialization

bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to

prevent thermal excitations

bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the

register

Relatively Long Decoherence Times

bull Mechanismsndash Radiative Decay ~10ps-1ns

bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations

bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral

bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps

or less

A Universal set of Quantum Gates

bull Single Qubit Rotations through laser induced Rabi Oscillations

bull CNOT operations through dipole interactions and laser excitation

Single Qubit Gates Rabi Flopping

bull Light-particle interaction is characterized by the product of the dipole moment and the electric field

μbullE(t)= ħR(t)

Where R(t) is the Rabi frequency and the pulse area is given by

Θ(t)=intR(t)dt

and the state at time t is then given by

Cos(Θ2)|0gt+Sin(Θ2)|1gt

Stufler et al

Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses

Cos(Θ2)|0gt+Sin(Θ2)|1gt

|1gt =gt electric charge

=gtPhotocurrent (PC)PC~Sin2(Θ2)

π-pulse corresponds to a population inversion

CNOT Dipole Coupling

Nearest neighbor interactions alter the energy states

Effective energy Ersquoi = Ei + sumjnei ∆Eij nj

Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied

Overcoming Short Interaction Distances

bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs

bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots

bull Optical Cavity coupling and fiber optical interconnects have also been proposed

Read Out of Specified Qubit States

bull Optical readoutExcitons decay spontaneously and the resulting radiation can be

measured

Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD

Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination

In either case repeated measurements have to be made A single shot readout still needs to be developed

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

• Quantum Dots
• Implementing Quantum Computers
• What is a quantum dot
• Gallium Arsenide Quantum Dots
• Energy Band Levels
• Slide 6
• Slide 7
• Tuning Quantum Dots
• Manufacturing methods
• Electron Beam Lithography
• Slide 11
• Molecular Beam Epitaxy
• Spin Quantum Computing
• Decoherence time ~ 100ns
• Artificial Atom
• Adding Electrons changing Vgate
• Comparison with Hydrogen
• Tuning the Quantum Dot
• Single Qubit manipulation
• Two Qubit Manipulation
• Ground State Splitting (J = Et ndash Es)
• Solving J(B(t)) Exchange Coupling
• SWAP - gate
• XOR ~ CNOT
• Read out Memory
• Another Spin up electron enters dot
• Spin down electron enters dot
• Slide 28
• 5 DiVincenzo QC Criteria
• The Physical System Excitons Trapped in GaAs Quantum Dots
• Initialization
• Relatively Long Decoherence Times
• A Universal set of Quantum Gates
• Single Qubit Gates Rabi Flopping
• Stufler et al
• CNOT Dipole Coupling
• Overcoming Short Interaction Distances
• Read Out of Specified Qubit States
• Slide 39

What is a quantum dot

bull In two words a semiconductor nanocrystal

bull Easily tunable by changing the size and composition of the nanocrystal

Gallium Arsenide Quantum Dots

bull Gallium arsenide is a III-V semiconductorndash Higher saturated electron velocity and higher

electron mobility than siliconndash Gallium arsenide can emit and absorb light

unlike siliconbull No silicon laser is possible (or has been made yet)

Energy Band Levels

bull Electrons exist in discrete energy levels in bulk semiconductor materialndash There exists a forbidden

range of energy levels in any material called the band gap

Energy Band Levels

bull By absorbing some sort of stimulus (in light or heat form) an electron can rise to the conduction band from the valence bandndash This action leaves behind a

ldquoholerdquo in the valence band The hole and the electron together are called an exciton

Energy Band Levels

bull The average distance between an electron and a hole in a exciton is called the Excited Bohr Radius

bull When the size of the semiconductor falls below the Bohr Radius the semiconductor is called a quantum dot

Tuning Quantum Dots

bull By changing size shape and composition quantum dots can change their absorptive and emissive properties dramatically

Manufacturing methods

bull Electron beam lithography

bull Molecular beam epitaxy

Electron Beam Lithography

bull Electrons are accelerated out of an electron gun and sent through condenser lens optics directly onto a wafer

bull λ = (123 Aring radicV)bull Advantages

ndash generation of micron and submicron resist geometries

ndash greater depth of focus than optical lithography

ndash masks are unnecessaryndash Optical diffraction limit is not a

real concern

Electron Beam Lithography

bull Disadvantage(s)ndash The lithography is serial

(masks arenrsquot used instead the beam itself sweeps across the wafer) =gt Comparatively low throughput ~5 wafers per hour at less than 1 micrometer resolution

ndash The proximity effect Electrons scatter because they are relatively low in mass reducing the resolution

bull Heavy ion lithography has been proposed but still is in development stages

Molecular Beam Epitaxy

bull Molecular beam epitaxy (MBE) is the deposition of one or more pure materials onto a single crystal wafer one layer of atoms at a time in order to form a perfect crystalndash This is done by evaporating each of the elements to

combine then condensing them on top of the waferndash The word ldquobeamrdquo means that the evaporated atoms

only meet each other on the wafer

Spin Quantum Computing

Qubit information is stored in the spin state of an electron in an artificial atom

AdvantagesLong decoherence time

Future Scalabilty

Artifical atoms are bigger than regular atoms therefore easier to manipulate

Decoherence time ~ 100ns

bull Time before the quantum mechanical system starts acting in a classical way with its complex environment

bull The state of the system has not yet collapsed due to (unwanted) environmental effects

bull Spin - DT are 100 as long as for the Excitonbull Need to SWITCH 104 during DT for reliable error

correction This requirement is met

Artificial Atombull Double Barrier

Heterostructurebull Dot In005Ga095Asbull Source ampDrain GaAsbull 2D Electron Gasbull Confine with gate biasbull D ~ Fermi wavelength

rarr Discrete energy levels

Adding Electrons changing Vgate

bull 2D-Harmonic Oscillator

bull Shell structure as in atoms

bull Magic Numbers 2 6 12

bull To add ldquoevenrdquo electron requires only additional Coulomb energy

Comparison with Hydrogenbull Artificial Atom

Energy levels ~ 1meV

Size ~ 10μm

Weak magnetic fields can affect energy levels

bull Hydrogen

Energy levels ~ 1eV

Size ~ 1Aring

Only strong magnetic fields can perturb energy levels

Factor 1000

Tuning the Quantum Dot

bull Tune so we have one valence electron

bull Initial state can be set by applying homogeneous magnetic field rarr |0gt

bull Low temperature kT lt ΔE (state gap)

bull Now we have defined our single qubit

Energy

position

Gate bias

Spin up - electron

Unoccupied state

Single Qubit manipulation

bull Unitary operations can be made by applying a local magnetic field H

ZE = -μB = g

μB SB

bull MF microscopebull AF microscopebull Sub grid of currentbull Magnetic dotsbull Etc

(Magnetic force microscope tip)

Two Qubit Manipulation

bull Complete set of logic requires a CNOT

bull Dots are placed so close that they overlap and interact

bull Hspin

= J(t)S1S

2

Exchange couplingJ(tEB) = E

triplet -E

singlet

(4th order Harmonic Oscillator)

Ground State Splitting (J = Et ndash E

s)

bull 2 coupled fermions must have an total anti-symmetric wave function

bull Lowest coupled state is the singlet It has a symmetric spatial wave function and an anti symmetric spin (Coulomb dominates)|ψ

sgt ~ (|12gt + |21gt) (|darruarrgt - |uarrdarrgt)

bull The triplet states are (|darrdarrgt)|ψ

tgt ~ (|12gt - |21gt) (|darruarrgt + |uarrdarrgt)

(|uarruarrgt)bull lt1|2gt ne 0 |igt is spatial wf Coulomb dominates

Solving J(B(t)) Exchange Coupling

bull Different solutions Heitler-London Hund-Mulliken Hubbard

bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations

SWAP - gatebull Assume J can be pulsed

J(t) = 0 J0

Formula 1

Formula 2

bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]

XOR ~ CNOT

bull Formula 3

bull Requirements Spin rotations about the z-axis Squareroot of U

swap

Read out Memory

bull Assume dot with an electron with some information stored in spin-state

bull Connect two leads to dot

bull Apply a small bias (ΔV) rarr Current (i)Energy

position

Gate bias

Spin up - electron

Unoccupied state

i

Another Spin up electron enters dot

bull Pauli principle forces electrons with spin up to occupy the higher energy state

bull Negligible chance of tunneling

E

position

Gate bias

Spin up - electron

Higher energy level(forbidden classically)

i=0

Spin down electron enters dot

bull Pauli principle allows the new electron to join the same energy level as the original electron

bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow

E

position

Gate bias

Spin up - electron

Unoccupied state

ine0

Read out Memory

bull We have a way of measuring the spin state of an electron in a quantum dot

bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties

bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique

bull The electron current can be specialized (we can aim its spin to make measurement efficient)

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

The Physical System Excitons Trapped in GaAs Quantum Dots

bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid

bull Often formed when photons excite electrons from the valence band into the conduction band

bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses

bull Decay by radiating photons Decay time ~50ps-1ns

bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt

Initialization

bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to

prevent thermal excitations

bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the

register

Relatively Long Decoherence Times

bull Mechanismsndash Radiative Decay ~10ps-1ns

bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations

bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral

bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps

or less

A Universal set of Quantum Gates

bull Single Qubit Rotations through laser induced Rabi Oscillations

bull CNOT operations through dipole interactions and laser excitation

Single Qubit Gates Rabi Flopping

bull Light-particle interaction is characterized by the product of the dipole moment and the electric field

μbullE(t)= ħR(t)

Where R(t) is the Rabi frequency and the pulse area is given by

Θ(t)=intR(t)dt

and the state at time t is then given by

Cos(Θ2)|0gt+Sin(Θ2)|1gt

Stufler et al

Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses

Cos(Θ2)|0gt+Sin(Θ2)|1gt

|1gt =gt electric charge

=gtPhotocurrent (PC)PC~Sin2(Θ2)

π-pulse corresponds to a population inversion

CNOT Dipole Coupling

Nearest neighbor interactions alter the energy states

Effective energy Ersquoi = Ei + sumjnei ∆Eij nj

Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied

Overcoming Short Interaction Distances

bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs

bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots

bull Optical Cavity coupling and fiber optical interconnects have also been proposed

Read Out of Specified Qubit States

bull Optical readoutExcitons decay spontaneously and the resulting radiation can be

measured

Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD

Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination

In either case repeated measurements have to be made A single shot readout still needs to be developed

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

• Quantum Dots
• Implementing Quantum Computers
• What is a quantum dot
• Gallium Arsenide Quantum Dots
• Energy Band Levels
• Slide 6
• Slide 7
• Tuning Quantum Dots
• Manufacturing methods
• Electron Beam Lithography
• Slide 11
• Molecular Beam Epitaxy
• Spin Quantum Computing
• Decoherence time ~ 100ns
• Artificial Atom
• Adding Electrons changing Vgate
• Comparison with Hydrogen
• Tuning the Quantum Dot
• Single Qubit manipulation
• Two Qubit Manipulation
• Ground State Splitting (J = Et ndash Es)
• Solving J(B(t)) Exchange Coupling
• SWAP - gate
• XOR ~ CNOT
• Read out Memory
• Another Spin up electron enters dot
• Spin down electron enters dot
• Slide 28
• 5 DiVincenzo QC Criteria
• The Physical System Excitons Trapped in GaAs Quantum Dots
• Initialization
• Relatively Long Decoherence Times
• A Universal set of Quantum Gates
• Single Qubit Gates Rabi Flopping
• Stufler et al
• CNOT Dipole Coupling
• Overcoming Short Interaction Distances
• Read Out of Specified Qubit States
• Slide 39

Gallium Arsenide Quantum Dots

bull Gallium arsenide is a III-V semiconductorndash Higher saturated electron velocity and higher

electron mobility than siliconndash Gallium arsenide can emit and absorb light

unlike siliconbull No silicon laser is possible (or has been made yet)

Energy Band Levels

bull Electrons exist in discrete energy levels in bulk semiconductor materialndash There exists a forbidden

range of energy levels in any material called the band gap

Energy Band Levels

bull By absorbing some sort of stimulus (in light or heat form) an electron can rise to the conduction band from the valence bandndash This action leaves behind a

ldquoholerdquo in the valence band The hole and the electron together are called an exciton

Energy Band Levels

bull The average distance between an electron and a hole in a exciton is called the Excited Bohr Radius

bull When the size of the semiconductor falls below the Bohr Radius the semiconductor is called a quantum dot

Tuning Quantum Dots

bull By changing size shape and composition quantum dots can change their absorptive and emissive properties dramatically

Manufacturing methods

bull Electron beam lithography

bull Molecular beam epitaxy

Electron Beam Lithography

bull Electrons are accelerated out of an electron gun and sent through condenser lens optics directly onto a wafer

bull λ = (123 Aring radicV)bull Advantages

ndash generation of micron and submicron resist geometries

ndash greater depth of focus than optical lithography

ndash masks are unnecessaryndash Optical diffraction limit is not a

real concern

Electron Beam Lithography

bull Disadvantage(s)ndash The lithography is serial

(masks arenrsquot used instead the beam itself sweeps across the wafer) =gt Comparatively low throughput ~5 wafers per hour at less than 1 micrometer resolution

ndash The proximity effect Electrons scatter because they are relatively low in mass reducing the resolution

bull Heavy ion lithography has been proposed but still is in development stages

Molecular Beam Epitaxy

bull Molecular beam epitaxy (MBE) is the deposition of one or more pure materials onto a single crystal wafer one layer of atoms at a time in order to form a perfect crystalndash This is done by evaporating each of the elements to

combine then condensing them on top of the waferndash The word ldquobeamrdquo means that the evaporated atoms

only meet each other on the wafer

Spin Quantum Computing

Qubit information is stored in the spin state of an electron in an artificial atom

AdvantagesLong decoherence time

Future Scalabilty

Artifical atoms are bigger than regular atoms therefore easier to manipulate

Decoherence time ~ 100ns

bull Time before the quantum mechanical system starts acting in a classical way with its complex environment

bull The state of the system has not yet collapsed due to (unwanted) environmental effects

bull Spin - DT are 100 as long as for the Excitonbull Need to SWITCH 104 during DT for reliable error

correction This requirement is met

Artificial Atombull Double Barrier

Heterostructurebull Dot In005Ga095Asbull Source ampDrain GaAsbull 2D Electron Gasbull Confine with gate biasbull D ~ Fermi wavelength

rarr Discrete energy levels

Adding Electrons changing Vgate

bull 2D-Harmonic Oscillator

bull Shell structure as in atoms

bull Magic Numbers 2 6 12

bull To add ldquoevenrdquo electron requires only additional Coulomb energy

Comparison with Hydrogenbull Artificial Atom

Energy levels ~ 1meV

Size ~ 10μm

Weak magnetic fields can affect energy levels

bull Hydrogen

Energy levels ~ 1eV

Size ~ 1Aring

Only strong magnetic fields can perturb energy levels

Factor 1000

Tuning the Quantum Dot

bull Tune so we have one valence electron

bull Initial state can be set by applying homogeneous magnetic field rarr |0gt

bull Low temperature kT lt ΔE (state gap)

bull Now we have defined our single qubit

Energy

position

Gate bias

Spin up - electron

Unoccupied state

Single Qubit manipulation

bull Unitary operations can be made by applying a local magnetic field H

ZE = -μB = g

μB SB

bull MF microscopebull AF microscopebull Sub grid of currentbull Magnetic dotsbull Etc

(Magnetic force microscope tip)

Two Qubit Manipulation

bull Complete set of logic requires a CNOT

bull Dots are placed so close that they overlap and interact

bull Hspin

= J(t)S1S

2

Exchange couplingJ(tEB) = E

triplet -E

singlet

(4th order Harmonic Oscillator)

Ground State Splitting (J = Et ndash E

s)

bull 2 coupled fermions must have an total anti-symmetric wave function

bull Lowest coupled state is the singlet It has a symmetric spatial wave function and an anti symmetric spin (Coulomb dominates)|ψ

sgt ~ (|12gt + |21gt) (|darruarrgt - |uarrdarrgt)

bull The triplet states are (|darrdarrgt)|ψ

tgt ~ (|12gt - |21gt) (|darruarrgt + |uarrdarrgt)

(|uarruarrgt)bull lt1|2gt ne 0 |igt is spatial wf Coulomb dominates

Solving J(B(t)) Exchange Coupling

bull Different solutions Heitler-London Hund-Mulliken Hubbard

bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations

SWAP - gatebull Assume J can be pulsed

J(t) = 0 J0

Formula 1

Formula 2

bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]

XOR ~ CNOT

bull Formula 3

bull Requirements Spin rotations about the z-axis Squareroot of U

swap

Read out Memory

bull Assume dot with an electron with some information stored in spin-state

bull Connect two leads to dot

bull Apply a small bias (ΔV) rarr Current (i)Energy

position

Gate bias

Spin up - electron

Unoccupied state

i

Another Spin up electron enters dot

bull Pauli principle forces electrons with spin up to occupy the higher energy state

bull Negligible chance of tunneling

E

position

Gate bias

Spin up - electron

Higher energy level(forbidden classically)

i=0

Spin down electron enters dot

bull Pauli principle allows the new electron to join the same energy level as the original electron

bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow

E

position

Gate bias

Spin up - electron

Unoccupied state

ine0

Read out Memory

bull We have a way of measuring the spin state of an electron in a quantum dot

bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties

bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique

bull The electron current can be specialized (we can aim its spin to make measurement efficient)

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

The Physical System Excitons Trapped in GaAs Quantum Dots

bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid

bull Often formed when photons excite electrons from the valence band into the conduction band

bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses

bull Decay by radiating photons Decay time ~50ps-1ns

bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt

Initialization

bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to

prevent thermal excitations

bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the

register

Relatively Long Decoherence Times

bull Mechanismsndash Radiative Decay ~10ps-1ns

bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations

bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral

bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps

or less

A Universal set of Quantum Gates

bull Single Qubit Rotations through laser induced Rabi Oscillations

bull CNOT operations through dipole interactions and laser excitation

Single Qubit Gates Rabi Flopping

bull Light-particle interaction is characterized by the product of the dipole moment and the electric field

μbullE(t)= ħR(t)

Where R(t) is the Rabi frequency and the pulse area is given by

Θ(t)=intR(t)dt

and the state at time t is then given by

Cos(Θ2)|0gt+Sin(Θ2)|1gt

Stufler et al

Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses

Cos(Θ2)|0gt+Sin(Θ2)|1gt

|1gt =gt electric charge

=gtPhotocurrent (PC)PC~Sin2(Θ2)

π-pulse corresponds to a population inversion

CNOT Dipole Coupling

Nearest neighbor interactions alter the energy states

Effective energy Ersquoi = Ei + sumjnei ∆Eij nj

Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied

Overcoming Short Interaction Distances

bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs

bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots

bull Optical Cavity coupling and fiber optical interconnects have also been proposed

Read Out of Specified Qubit States

bull Optical readoutExcitons decay spontaneously and the resulting radiation can be

measured

Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD

Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination

In either case repeated measurements have to be made A single shot readout still needs to be developed

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

• Quantum Dots
• Implementing Quantum Computers
• What is a quantum dot
• Gallium Arsenide Quantum Dots
• Energy Band Levels
• Slide 6
• Slide 7
• Tuning Quantum Dots
• Manufacturing methods
• Electron Beam Lithography
• Slide 11
• Molecular Beam Epitaxy
• Spin Quantum Computing
• Decoherence time ~ 100ns
• Artificial Atom
• Adding Electrons changing Vgate
• Comparison with Hydrogen
• Tuning the Quantum Dot
• Single Qubit manipulation
• Two Qubit Manipulation
• Ground State Splitting (J = Et ndash Es)
• Solving J(B(t)) Exchange Coupling
• SWAP - gate
• XOR ~ CNOT
• Read out Memory
• Another Spin up electron enters dot
• Spin down electron enters dot
• Slide 28
• 5 DiVincenzo QC Criteria
• The Physical System Excitons Trapped in GaAs Quantum Dots
• Initialization
• Relatively Long Decoherence Times
• A Universal set of Quantum Gates
• Single Qubit Gates Rabi Flopping
• Stufler et al
• CNOT Dipole Coupling
• Overcoming Short Interaction Distances
• Read Out of Specified Qubit States
• Slide 39

Energy Band Levels

bull Electrons exist in discrete energy levels in bulk semiconductor materialndash There exists a forbidden

range of energy levels in any material called the band gap

Energy Band Levels

bull By absorbing some sort of stimulus (in light or heat form) an electron can rise to the conduction band from the valence bandndash This action leaves behind a

ldquoholerdquo in the valence band The hole and the electron together are called an exciton

Energy Band Levels

bull The average distance between an electron and a hole in a exciton is called the Excited Bohr Radius

bull When the size of the semiconductor falls below the Bohr Radius the semiconductor is called a quantum dot

Tuning Quantum Dots

bull By changing size shape and composition quantum dots can change their absorptive and emissive properties dramatically

Manufacturing methods

bull Electron beam lithography

bull Molecular beam epitaxy

Electron Beam Lithography

bull Electrons are accelerated out of an electron gun and sent through condenser lens optics directly onto a wafer

bull λ = (123 Aring radicV)bull Advantages

ndash generation of micron and submicron resist geometries

ndash greater depth of focus than optical lithography

ndash masks are unnecessaryndash Optical diffraction limit is not a

real concern

Electron Beam Lithography

bull Disadvantage(s)ndash The lithography is serial

(masks arenrsquot used instead the beam itself sweeps across the wafer) =gt Comparatively low throughput ~5 wafers per hour at less than 1 micrometer resolution

ndash The proximity effect Electrons scatter because they are relatively low in mass reducing the resolution

bull Heavy ion lithography has been proposed but still is in development stages

Molecular Beam Epitaxy

bull Molecular beam epitaxy (MBE) is the deposition of one or more pure materials onto a single crystal wafer one layer of atoms at a time in order to form a perfect crystalndash This is done by evaporating each of the elements to

combine then condensing them on top of the waferndash The word ldquobeamrdquo means that the evaporated atoms

only meet each other on the wafer

Spin Quantum Computing

Qubit information is stored in the spin state of an electron in an artificial atom

AdvantagesLong decoherence time

Future Scalabilty

Artifical atoms are bigger than regular atoms therefore easier to manipulate

Decoherence time ~ 100ns

bull Time before the quantum mechanical system starts acting in a classical way with its complex environment

bull The state of the system has not yet collapsed due to (unwanted) environmental effects

bull Spin - DT are 100 as long as for the Excitonbull Need to SWITCH 104 during DT for reliable error

correction This requirement is met

Artificial Atombull Double Barrier

Heterostructurebull Dot In005Ga095Asbull Source ampDrain GaAsbull 2D Electron Gasbull Confine with gate biasbull D ~ Fermi wavelength

rarr Discrete energy levels

Adding Electrons changing Vgate

bull 2D-Harmonic Oscillator

bull Shell structure as in atoms

bull Magic Numbers 2 6 12

bull To add ldquoevenrdquo electron requires only additional Coulomb energy

Comparison with Hydrogenbull Artificial Atom

Energy levels ~ 1meV

Size ~ 10μm

Weak magnetic fields can affect energy levels

bull Hydrogen

Energy levels ~ 1eV

Size ~ 1Aring

Only strong magnetic fields can perturb energy levels

Factor 1000

Tuning the Quantum Dot

bull Tune so we have one valence electron

bull Initial state can be set by applying homogeneous magnetic field rarr |0gt

bull Low temperature kT lt ΔE (state gap)

bull Now we have defined our single qubit

Energy

position

Gate bias

Spin up - electron

Unoccupied state

Single Qubit manipulation

bull Unitary operations can be made by applying a local magnetic field H

ZE = -μB = g

μB SB

bull MF microscopebull AF microscopebull Sub grid of currentbull Magnetic dotsbull Etc

(Magnetic force microscope tip)

Two Qubit Manipulation

bull Complete set of logic requires a CNOT

bull Dots are placed so close that they overlap and interact

bull Hspin

= J(t)S1S

2

Exchange couplingJ(tEB) = E

triplet -E

singlet

(4th order Harmonic Oscillator)

Ground State Splitting (J = Et ndash E

s)

bull 2 coupled fermions must have an total anti-symmetric wave function

bull Lowest coupled state is the singlet It has a symmetric spatial wave function and an anti symmetric spin (Coulomb dominates)|ψ

sgt ~ (|12gt + |21gt) (|darruarrgt - |uarrdarrgt)

bull The triplet states are (|darrdarrgt)|ψ

tgt ~ (|12gt - |21gt) (|darruarrgt + |uarrdarrgt)

(|uarruarrgt)bull lt1|2gt ne 0 |igt is spatial wf Coulomb dominates

Solving J(B(t)) Exchange Coupling

bull Different solutions Heitler-London Hund-Mulliken Hubbard

bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations

SWAP - gatebull Assume J can be pulsed

J(t) = 0 J0

Formula 1

Formula 2

bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]

XOR ~ CNOT

bull Formula 3

bull Requirements Spin rotations about the z-axis Squareroot of U

swap

Read out Memory

bull Assume dot with an electron with some information stored in spin-state

bull Connect two leads to dot

bull Apply a small bias (ΔV) rarr Current (i)Energy

position

Gate bias

Spin up - electron

Unoccupied state

i

Another Spin up electron enters dot

bull Pauli principle forces electrons with spin up to occupy the higher energy state

bull Negligible chance of tunneling

E

position

Gate bias

Spin up - electron

Higher energy level(forbidden classically)

i=0

Spin down electron enters dot

bull Pauli principle allows the new electron to join the same energy level as the original electron

bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow

E

position

Gate bias

Spin up - electron

Unoccupied state

ine0

Read out Memory

bull We have a way of measuring the spin state of an electron in a quantum dot

bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties

bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique

bull The electron current can be specialized (we can aim its spin to make measurement efficient)

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

The Physical System Excitons Trapped in GaAs Quantum Dots

bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid

bull Often formed when photons excite electrons from the valence band into the conduction band

bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses

bull Decay by radiating photons Decay time ~50ps-1ns

bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt

Initialization

bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to

prevent thermal excitations

bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the

register

Relatively Long Decoherence Times

bull Mechanismsndash Radiative Decay ~10ps-1ns

bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations

bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral

bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps

or less

A Universal set of Quantum Gates

bull Single Qubit Rotations through laser induced Rabi Oscillations

bull CNOT operations through dipole interactions and laser excitation

Single Qubit Gates Rabi Flopping

bull Light-particle interaction is characterized by the product of the dipole moment and the electric field

μbullE(t)= ħR(t)

Where R(t) is the Rabi frequency and the pulse area is given by

Θ(t)=intR(t)dt

and the state at time t is then given by

Cos(Θ2)|0gt+Sin(Θ2)|1gt

Stufler et al

Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses

Cos(Θ2)|0gt+Sin(Θ2)|1gt

|1gt =gt electric charge

=gtPhotocurrent (PC)PC~Sin2(Θ2)

π-pulse corresponds to a population inversion

CNOT Dipole Coupling

Nearest neighbor interactions alter the energy states

Effective energy Ersquoi = Ei + sumjnei ∆Eij nj

Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied

Overcoming Short Interaction Distances

bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs

bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots

bull Optical Cavity coupling and fiber optical interconnects have also been proposed

Read Out of Specified Qubit States

bull Optical readoutExcitons decay spontaneously and the resulting radiation can be

measured

Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD

Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination

In either case repeated measurements have to be made A single shot readout still needs to be developed

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

• Quantum Dots
• Implementing Quantum Computers
• What is a quantum dot
• Gallium Arsenide Quantum Dots
• Energy Band Levels
• Slide 6
• Slide 7
• Tuning Quantum Dots
• Manufacturing methods
• Electron Beam Lithography
• Slide 11
• Molecular Beam Epitaxy
• Spin Quantum Computing
• Decoherence time ~ 100ns
• Artificial Atom
• Adding Electrons changing Vgate
• Comparison with Hydrogen
• Tuning the Quantum Dot
• Single Qubit manipulation
• Two Qubit Manipulation
• Ground State Splitting (J = Et ndash Es)
• Solving J(B(t)) Exchange Coupling
• SWAP - gate
• XOR ~ CNOT
• Read out Memory
• Another Spin up electron enters dot
• Spin down electron enters dot
• Slide 28
• 5 DiVincenzo QC Criteria
• The Physical System Excitons Trapped in GaAs Quantum Dots
• Initialization
• Relatively Long Decoherence Times
• A Universal set of Quantum Gates
• Single Qubit Gates Rabi Flopping
• Stufler et al
• CNOT Dipole Coupling
• Overcoming Short Interaction Distances
• Read Out of Specified Qubit States
• Slide 39

Energy Band Levels

bull By absorbing some sort of stimulus (in light or heat form) an electron can rise to the conduction band from the valence bandndash This action leaves behind a

ldquoholerdquo in the valence band The hole and the electron together are called an exciton

Energy Band Levels

bull The average distance between an electron and a hole in a exciton is called the Excited Bohr Radius

bull When the size of the semiconductor falls below the Bohr Radius the semiconductor is called a quantum dot

Tuning Quantum Dots

bull By changing size shape and composition quantum dots can change their absorptive and emissive properties dramatically

Manufacturing methods

bull Electron beam lithography

bull Molecular beam epitaxy

Electron Beam Lithography

bull Electrons are accelerated out of an electron gun and sent through condenser lens optics directly onto a wafer

bull λ = (123 Aring radicV)bull Advantages

ndash generation of micron and submicron resist geometries

ndash greater depth of focus than optical lithography

ndash masks are unnecessaryndash Optical diffraction limit is not a

real concern

Electron Beam Lithography

bull Disadvantage(s)ndash The lithography is serial

(masks arenrsquot used instead the beam itself sweeps across the wafer) =gt Comparatively low throughput ~5 wafers per hour at less than 1 micrometer resolution

ndash The proximity effect Electrons scatter because they are relatively low in mass reducing the resolution

bull Heavy ion lithography has been proposed but still is in development stages

Molecular Beam Epitaxy

bull Molecular beam epitaxy (MBE) is the deposition of one or more pure materials onto a single crystal wafer one layer of atoms at a time in order to form a perfect crystalndash This is done by evaporating each of the elements to

combine then condensing them on top of the waferndash The word ldquobeamrdquo means that the evaporated atoms

only meet each other on the wafer

Spin Quantum Computing

Qubit information is stored in the spin state of an electron in an artificial atom

AdvantagesLong decoherence time

Future Scalabilty

Artifical atoms are bigger than regular atoms therefore easier to manipulate

Decoherence time ~ 100ns

bull Time before the quantum mechanical system starts acting in a classical way with its complex environment

bull The state of the system has not yet collapsed due to (unwanted) environmental effects

bull Spin - DT are 100 as long as for the Excitonbull Need to SWITCH 104 during DT for reliable error

correction This requirement is met

Artificial Atombull Double Barrier

Heterostructurebull Dot In005Ga095Asbull Source ampDrain GaAsbull 2D Electron Gasbull Confine with gate biasbull D ~ Fermi wavelength

rarr Discrete energy levels

Adding Electrons changing Vgate

bull 2D-Harmonic Oscillator

bull Shell structure as in atoms

bull Magic Numbers 2 6 12

bull To add ldquoevenrdquo electron requires only additional Coulomb energy

Comparison with Hydrogenbull Artificial Atom

Energy levels ~ 1meV

Size ~ 10μm

Weak magnetic fields can affect energy levels

bull Hydrogen

Energy levels ~ 1eV

Size ~ 1Aring

Only strong magnetic fields can perturb energy levels

Factor 1000

Tuning the Quantum Dot

bull Tune so we have one valence electron

bull Initial state can be set by applying homogeneous magnetic field rarr |0gt

bull Low temperature kT lt ΔE (state gap)

bull Now we have defined our single qubit

Energy

position

Gate bias

Spin up - electron

Unoccupied state

Single Qubit manipulation

bull Unitary operations can be made by applying a local magnetic field H

ZE = -μB = g

μB SB

bull MF microscopebull AF microscopebull Sub grid of currentbull Magnetic dotsbull Etc

(Magnetic force microscope tip)

Two Qubit Manipulation

bull Complete set of logic requires a CNOT

bull Dots are placed so close that they overlap and interact

bull Hspin

= J(t)S1S

2

Exchange couplingJ(tEB) = E

triplet -E

singlet

(4th order Harmonic Oscillator)

Ground State Splitting (J = Et ndash E

s)

bull 2 coupled fermions must have an total anti-symmetric wave function

bull Lowest coupled state is the singlet It has a symmetric spatial wave function and an anti symmetric spin (Coulomb dominates)|ψ

sgt ~ (|12gt + |21gt) (|darruarrgt - |uarrdarrgt)

bull The triplet states are (|darrdarrgt)|ψ

tgt ~ (|12gt - |21gt) (|darruarrgt + |uarrdarrgt)

(|uarruarrgt)bull lt1|2gt ne 0 |igt is spatial wf Coulomb dominates

Solving J(B(t)) Exchange Coupling

bull Different solutions Heitler-London Hund-Mulliken Hubbard

bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations

SWAP - gatebull Assume J can be pulsed

J(t) = 0 J0

Formula 1

Formula 2

bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]

XOR ~ CNOT

bull Formula 3

bull Requirements Spin rotations about the z-axis Squareroot of U

swap

Read out Memory

bull Assume dot with an electron with some information stored in spin-state

bull Connect two leads to dot

bull Apply a small bias (ΔV) rarr Current (i)Energy

position

Gate bias

Spin up - electron

Unoccupied state

i

Another Spin up electron enters dot

bull Pauli principle forces electrons with spin up to occupy the higher energy state

bull Negligible chance of tunneling

E

position

Gate bias

Spin up - electron

Higher energy level(forbidden classically)

i=0

Spin down electron enters dot

bull Pauli principle allows the new electron to join the same energy level as the original electron

bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow

E

position

Gate bias

Spin up - electron

Unoccupied state

ine0

Read out Memory

bull We have a way of measuring the spin state of an electron in a quantum dot

bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties

bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique

bull The electron current can be specialized (we can aim its spin to make measurement efficient)

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

The Physical System Excitons Trapped in GaAs Quantum Dots

bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid

bull Often formed when photons excite electrons from the valence band into the conduction band

bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses

bull Decay by radiating photons Decay time ~50ps-1ns

bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt

Initialization

bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to

prevent thermal excitations

bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the

register

Relatively Long Decoherence Times

bull Mechanismsndash Radiative Decay ~10ps-1ns

bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations

bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral

bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps

or less

A Universal set of Quantum Gates

bull Single Qubit Rotations through laser induced Rabi Oscillations

bull CNOT operations through dipole interactions and laser excitation

Single Qubit Gates Rabi Flopping

bull Light-particle interaction is characterized by the product of the dipole moment and the electric field

μbullE(t)= ħR(t)

Where R(t) is the Rabi frequency and the pulse area is given by

Θ(t)=intR(t)dt

and the state at time t is then given by

Cos(Θ2)|0gt+Sin(Θ2)|1gt

Stufler et al

Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses

Cos(Θ2)|0gt+Sin(Θ2)|1gt

|1gt =gt electric charge

=gtPhotocurrent (PC)PC~Sin2(Θ2)

π-pulse corresponds to a population inversion

CNOT Dipole Coupling

Nearest neighbor interactions alter the energy states

Effective energy Ersquoi = Ei + sumjnei ∆Eij nj

Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied

Overcoming Short Interaction Distances

bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs

bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots

bull Optical Cavity coupling and fiber optical interconnects have also been proposed

Read Out of Specified Qubit States

bull Optical readoutExcitons decay spontaneously and the resulting radiation can be

measured

Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD

Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination

In either case repeated measurements have to be made A single shot readout still needs to be developed

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

• Quantum Dots
• Implementing Quantum Computers
• What is a quantum dot
• Gallium Arsenide Quantum Dots
• Energy Band Levels
• Slide 6
• Slide 7
• Tuning Quantum Dots
• Manufacturing methods
• Electron Beam Lithography
• Slide 11
• Molecular Beam Epitaxy
• Spin Quantum Computing
• Decoherence time ~ 100ns
• Artificial Atom
• Adding Electrons changing Vgate
• Comparison with Hydrogen
• Tuning the Quantum Dot
• Single Qubit manipulation
• Two Qubit Manipulation
• Ground State Splitting (J = Et ndash Es)
• Solving J(B(t)) Exchange Coupling
• SWAP - gate
• XOR ~ CNOT
• Read out Memory
• Another Spin up electron enters dot
• Spin down electron enters dot
• Slide 28
• 5 DiVincenzo QC Criteria
• The Physical System Excitons Trapped in GaAs Quantum Dots
• Initialization
• Relatively Long Decoherence Times
• A Universal set of Quantum Gates
• Single Qubit Gates Rabi Flopping
• Stufler et al
• CNOT Dipole Coupling
• Overcoming Short Interaction Distances
• Read Out of Specified Qubit States
• Slide 39

Energy Band Levels

bull The average distance between an electron and a hole in a exciton is called the Excited Bohr Radius

bull When the size of the semiconductor falls below the Bohr Radius the semiconductor is called a quantum dot

Tuning Quantum Dots

bull By changing size shape and composition quantum dots can change their absorptive and emissive properties dramatically

Manufacturing methods

bull Electron beam lithography

bull Molecular beam epitaxy

Electron Beam Lithography

bull Electrons are accelerated out of an electron gun and sent through condenser lens optics directly onto a wafer

bull λ = (123 Aring radicV)bull Advantages

ndash generation of micron and submicron resist geometries

ndash greater depth of focus than optical lithography

ndash masks are unnecessaryndash Optical diffraction limit is not a

real concern

Electron Beam Lithography

bull Disadvantage(s)ndash The lithography is serial

(masks arenrsquot used instead the beam itself sweeps across the wafer) =gt Comparatively low throughput ~5 wafers per hour at less than 1 micrometer resolution

ndash The proximity effect Electrons scatter because they are relatively low in mass reducing the resolution

bull Heavy ion lithography has been proposed but still is in development stages

Molecular Beam Epitaxy

bull Molecular beam epitaxy (MBE) is the deposition of one or more pure materials onto a single crystal wafer one layer of atoms at a time in order to form a perfect crystalndash This is done by evaporating each of the elements to

combine then condensing them on top of the waferndash The word ldquobeamrdquo means that the evaporated atoms

only meet each other on the wafer

Spin Quantum Computing

Qubit information is stored in the spin state of an electron in an artificial atom

AdvantagesLong decoherence time

Future Scalabilty

Artifical atoms are bigger than regular atoms therefore easier to manipulate

Decoherence time ~ 100ns

bull Time before the quantum mechanical system starts acting in a classical way with its complex environment

bull The state of the system has not yet collapsed due to (unwanted) environmental effects

bull Spin - DT are 100 as long as for the Excitonbull Need to SWITCH 104 during DT for reliable error

correction This requirement is met

Artificial Atombull Double Barrier

Heterostructurebull Dot In005Ga095Asbull Source ampDrain GaAsbull 2D Electron Gasbull Confine with gate biasbull D ~ Fermi wavelength

rarr Discrete energy levels

Adding Electrons changing Vgate

bull 2D-Harmonic Oscillator

bull Shell structure as in atoms

bull Magic Numbers 2 6 12

bull To add ldquoevenrdquo electron requires only additional Coulomb energy

Comparison with Hydrogenbull Artificial Atom

Energy levels ~ 1meV

Size ~ 10μm

Weak magnetic fields can affect energy levels

bull Hydrogen

Energy levels ~ 1eV

Size ~ 1Aring

Only strong magnetic fields can perturb energy levels

Factor 1000

Tuning the Quantum Dot

bull Tune so we have one valence electron

bull Initial state can be set by applying homogeneous magnetic field rarr |0gt

bull Low temperature kT lt ΔE (state gap)

bull Now we have defined our single qubit

Energy

position

Gate bias

Spin up - electron

Unoccupied state

Single Qubit manipulation

bull Unitary operations can be made by applying a local magnetic field H

ZE = -μB = g

μB SB

bull MF microscopebull AF microscopebull Sub grid of currentbull Magnetic dotsbull Etc

(Magnetic force microscope tip)

Two Qubit Manipulation

bull Complete set of logic requires a CNOT

bull Dots are placed so close that they overlap and interact

bull Hspin

= J(t)S1S

2

Exchange couplingJ(tEB) = E

triplet -E

singlet

(4th order Harmonic Oscillator)

Ground State Splitting (J = Et ndash E

s)

bull 2 coupled fermions must have an total anti-symmetric wave function

bull Lowest coupled state is the singlet It has a symmetric spatial wave function and an anti symmetric spin (Coulomb dominates)|ψ

sgt ~ (|12gt + |21gt) (|darruarrgt - |uarrdarrgt)

bull The triplet states are (|darrdarrgt)|ψ

tgt ~ (|12gt - |21gt) (|darruarrgt + |uarrdarrgt)

(|uarruarrgt)bull lt1|2gt ne 0 |igt is spatial wf Coulomb dominates

Solving J(B(t)) Exchange Coupling

bull Different solutions Heitler-London Hund-Mulliken Hubbard

bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations

SWAP - gatebull Assume J can be pulsed

J(t) = 0 J0

Formula 1

Formula 2

bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]

XOR ~ CNOT

bull Formula 3

bull Requirements Spin rotations about the z-axis Squareroot of U

swap

Read out Memory

bull Assume dot with an electron with some information stored in spin-state

bull Connect two leads to dot

bull Apply a small bias (ΔV) rarr Current (i)Energy

position

Gate bias

Spin up - electron

Unoccupied state

i

Another Spin up electron enters dot

bull Pauli principle forces electrons with spin up to occupy the higher energy state

bull Negligible chance of tunneling

E

position

Gate bias

Spin up - electron

Higher energy level(forbidden classically)

i=0

Spin down electron enters dot

bull Pauli principle allows the new electron to join the same energy level as the original electron

bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow

E

position

Gate bias

Spin up - electron

Unoccupied state

ine0

Read out Memory

bull We have a way of measuring the spin state of an electron in a quantum dot

bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties

bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique

bull The electron current can be specialized (we can aim its spin to make measurement efficient)

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

The Physical System Excitons Trapped in GaAs Quantum Dots

bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid

bull Often formed when photons excite electrons from the valence band into the conduction band

bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses

bull Decay by radiating photons Decay time ~50ps-1ns

bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt

Initialization

bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to

prevent thermal excitations

bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the

register

Relatively Long Decoherence Times

bull Mechanismsndash Radiative Decay ~10ps-1ns

bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations

bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral

bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps

or less

A Universal set of Quantum Gates

bull Single Qubit Rotations through laser induced Rabi Oscillations

bull CNOT operations through dipole interactions and laser excitation

Single Qubit Gates Rabi Flopping

bull Light-particle interaction is characterized by the product of the dipole moment and the electric field

μbullE(t)= ħR(t)

Where R(t) is the Rabi frequency and the pulse area is given by

Θ(t)=intR(t)dt

and the state at time t is then given by

Cos(Θ2)|0gt+Sin(Θ2)|1gt

Stufler et al

Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses

Cos(Θ2)|0gt+Sin(Θ2)|1gt

|1gt =gt electric charge

=gtPhotocurrent (PC)PC~Sin2(Θ2)

π-pulse corresponds to a population inversion

CNOT Dipole Coupling

Nearest neighbor interactions alter the energy states

Effective energy Ersquoi = Ei + sumjnei ∆Eij nj

Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied

Overcoming Short Interaction Distances

bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs

bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots

bull Optical Cavity coupling and fiber optical interconnects have also been proposed

Read Out of Specified Qubit States

bull Optical readoutExcitons decay spontaneously and the resulting radiation can be

measured

Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD

Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination

In either case repeated measurements have to be made A single shot readout still needs to be developed

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

• Quantum Dots
• Implementing Quantum Computers
• What is a quantum dot
• Gallium Arsenide Quantum Dots
• Energy Band Levels
• Slide 6
• Slide 7
• Tuning Quantum Dots
• Manufacturing methods
• Electron Beam Lithography
• Slide 11
• Molecular Beam Epitaxy
• Spin Quantum Computing
• Decoherence time ~ 100ns
• Artificial Atom
• Adding Electrons changing Vgate
• Comparison with Hydrogen
• Tuning the Quantum Dot
• Single Qubit manipulation
• Two Qubit Manipulation
• Ground State Splitting (J = Et ndash Es)
• Solving J(B(t)) Exchange Coupling
• SWAP - gate
• XOR ~ CNOT
• Read out Memory
• Another Spin up electron enters dot
• Spin down electron enters dot
• Slide 28
• 5 DiVincenzo QC Criteria
• The Physical System Excitons Trapped in GaAs Quantum Dots
• Initialization
• Relatively Long Decoherence Times
• A Universal set of Quantum Gates
• Single Qubit Gates Rabi Flopping
• Stufler et al
• CNOT Dipole Coupling
• Overcoming Short Interaction Distances
• Read Out of Specified Qubit States
• Slide 39

Tuning Quantum Dots

bull By changing size shape and composition quantum dots can change their absorptive and emissive properties dramatically

Manufacturing methods

bull Electron beam lithography

bull Molecular beam epitaxy

Electron Beam Lithography

bull Electrons are accelerated out of an electron gun and sent through condenser lens optics directly onto a wafer

bull λ = (123 Aring radicV)bull Advantages

ndash generation of micron and submicron resist geometries

ndash greater depth of focus than optical lithography

ndash masks are unnecessaryndash Optical diffraction limit is not a

real concern

Electron Beam Lithography

bull Disadvantage(s)ndash The lithography is serial

(masks arenrsquot used instead the beam itself sweeps across the wafer) =gt Comparatively low throughput ~5 wafers per hour at less than 1 micrometer resolution

ndash The proximity effect Electrons scatter because they are relatively low in mass reducing the resolution

bull Heavy ion lithography has been proposed but still is in development stages

Molecular Beam Epitaxy

bull Molecular beam epitaxy (MBE) is the deposition of one or more pure materials onto a single crystal wafer one layer of atoms at a time in order to form a perfect crystalndash This is done by evaporating each of the elements to

combine then condensing them on top of the waferndash The word ldquobeamrdquo means that the evaporated atoms

only meet each other on the wafer

Spin Quantum Computing

Qubit information is stored in the spin state of an electron in an artificial atom

AdvantagesLong decoherence time

Future Scalabilty

Artifical atoms are bigger than regular atoms therefore easier to manipulate

Decoherence time ~ 100ns

bull Time before the quantum mechanical system starts acting in a classical way with its complex environment

bull The state of the system has not yet collapsed due to (unwanted) environmental effects

bull Spin - DT are 100 as long as for the Excitonbull Need to SWITCH 104 during DT for reliable error

correction This requirement is met

Artificial Atombull Double Barrier

Heterostructurebull Dot In005Ga095Asbull Source ampDrain GaAsbull 2D Electron Gasbull Confine with gate biasbull D ~ Fermi wavelength

rarr Discrete energy levels

Adding Electrons changing Vgate

bull 2D-Harmonic Oscillator

bull Shell structure as in atoms

bull Magic Numbers 2 6 12

bull To add ldquoevenrdquo electron requires only additional Coulomb energy

Comparison with Hydrogenbull Artificial Atom

Energy levels ~ 1meV

Size ~ 10μm

Weak magnetic fields can affect energy levels

bull Hydrogen

Energy levels ~ 1eV

Size ~ 1Aring

Only strong magnetic fields can perturb energy levels

Factor 1000

Tuning the Quantum Dot

bull Tune so we have one valence electron

bull Initial state can be set by applying homogeneous magnetic field rarr |0gt

bull Low temperature kT lt ΔE (state gap)

bull Now we have defined our single qubit

Energy

position

Gate bias

Spin up - electron

Unoccupied state

Single Qubit manipulation

bull Unitary operations can be made by applying a local magnetic field H

ZE = -μB = g

μB SB

bull MF microscopebull AF microscopebull Sub grid of currentbull Magnetic dotsbull Etc

(Magnetic force microscope tip)

Two Qubit Manipulation

bull Complete set of logic requires a CNOT

bull Dots are placed so close that they overlap and interact

bull Hspin

= J(t)S1S

2

Exchange couplingJ(tEB) = E

triplet -E

singlet

(4th order Harmonic Oscillator)

Ground State Splitting (J = Et ndash E

s)

bull 2 coupled fermions must have an total anti-symmetric wave function

bull Lowest coupled state is the singlet It has a symmetric spatial wave function and an anti symmetric spin (Coulomb dominates)|ψ

sgt ~ (|12gt + |21gt) (|darruarrgt - |uarrdarrgt)

bull The triplet states are (|darrdarrgt)|ψ

tgt ~ (|12gt - |21gt) (|darruarrgt + |uarrdarrgt)

(|uarruarrgt)bull lt1|2gt ne 0 |igt is spatial wf Coulomb dominates

Solving J(B(t)) Exchange Coupling

bull Different solutions Heitler-London Hund-Mulliken Hubbard

bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations

SWAP - gatebull Assume J can be pulsed

J(t) = 0 J0

Formula 1

Formula 2

bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]

XOR ~ CNOT

bull Formula 3

bull Requirements Spin rotations about the z-axis Squareroot of U

swap

Read out Memory

bull Assume dot with an electron with some information stored in spin-state

bull Connect two leads to dot

bull Apply a small bias (ΔV) rarr Current (i)Energy

position

Gate bias

Spin up - electron

Unoccupied state

i

Another Spin up electron enters dot

bull Pauli principle forces electrons with spin up to occupy the higher energy state

bull Negligible chance of tunneling

E

position

Gate bias

Spin up - electron

Higher energy level(forbidden classically)

i=0

Spin down electron enters dot

bull Pauli principle allows the new electron to join the same energy level as the original electron

bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow

E

position

Gate bias

Spin up - electron

Unoccupied state

ine0

Read out Memory

bull We have a way of measuring the spin state of an electron in a quantum dot

bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties

bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique

bull The electron current can be specialized (we can aim its spin to make measurement efficient)

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

The Physical System Excitons Trapped in GaAs Quantum Dots

bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid

bull Often formed when photons excite electrons from the valence band into the conduction band

bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses

bull Decay by radiating photons Decay time ~50ps-1ns

bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt

Initialization

bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to

prevent thermal excitations

bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the

register

Relatively Long Decoherence Times

bull Mechanismsndash Radiative Decay ~10ps-1ns

bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations

bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral

bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps

or less

A Universal set of Quantum Gates

bull Single Qubit Rotations through laser induced Rabi Oscillations

bull CNOT operations through dipole interactions and laser excitation

Single Qubit Gates Rabi Flopping

bull Light-particle interaction is characterized by the product of the dipole moment and the electric field

μbullE(t)= ħR(t)

Where R(t) is the Rabi frequency and the pulse area is given by

Θ(t)=intR(t)dt

and the state at time t is then given by

Cos(Θ2)|0gt+Sin(Θ2)|1gt

Stufler et al

Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses

Cos(Θ2)|0gt+Sin(Θ2)|1gt

|1gt =gt electric charge

=gtPhotocurrent (PC)PC~Sin2(Θ2)

π-pulse corresponds to a population inversion

CNOT Dipole Coupling

Nearest neighbor interactions alter the energy states

Effective energy Ersquoi = Ei + sumjnei ∆Eij nj

Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied

Overcoming Short Interaction Distances

bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs

bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots

bull Optical Cavity coupling and fiber optical interconnects have also been proposed

Read Out of Specified Qubit States

bull Optical readoutExcitons decay spontaneously and the resulting radiation can be

measured

Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD

Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination

In either case repeated measurements have to be made A single shot readout still needs to be developed

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

• Quantum Dots
• Implementing Quantum Computers
• What is a quantum dot
• Gallium Arsenide Quantum Dots
• Energy Band Levels
• Slide 6
• Slide 7
• Tuning Quantum Dots
• Manufacturing methods
• Electron Beam Lithography
• Slide 11
• Molecular Beam Epitaxy
• Spin Quantum Computing
• Decoherence time ~ 100ns
• Artificial Atom
• Adding Electrons changing Vgate
• Comparison with Hydrogen
• Tuning the Quantum Dot
• Single Qubit manipulation
• Two Qubit Manipulation
• Ground State Splitting (J = Et ndash Es)
• Solving J(B(t)) Exchange Coupling
• SWAP - gate
• XOR ~ CNOT
• Read out Memory
• Another Spin up electron enters dot
• Spin down electron enters dot
• Slide 28
• 5 DiVincenzo QC Criteria
• The Physical System Excitons Trapped in GaAs Quantum Dots
• Initialization
• Relatively Long Decoherence Times
• A Universal set of Quantum Gates
• Single Qubit Gates Rabi Flopping
• Stufler et al
• CNOT Dipole Coupling
• Overcoming Short Interaction Distances
• Read Out of Specified Qubit States
• Slide 39

Manufacturing methods

bull Electron beam lithography

bull Molecular beam epitaxy

Electron Beam Lithography

bull Electrons are accelerated out of an electron gun and sent through condenser lens optics directly onto a wafer

bull λ = (123 Aring radicV)bull Advantages

ndash generation of micron and submicron resist geometries

ndash greater depth of focus than optical lithography

ndash masks are unnecessaryndash Optical diffraction limit is not a

real concern

Electron Beam Lithography

bull Disadvantage(s)ndash The lithography is serial

(masks arenrsquot used instead the beam itself sweeps across the wafer) =gt Comparatively low throughput ~5 wafers per hour at less than 1 micrometer resolution

ndash The proximity effect Electrons scatter because they are relatively low in mass reducing the resolution

bull Heavy ion lithography has been proposed but still is in development stages

Molecular Beam Epitaxy

bull Molecular beam epitaxy (MBE) is the deposition of one or more pure materials onto a single crystal wafer one layer of atoms at a time in order to form a perfect crystalndash This is done by evaporating each of the elements to

combine then condensing them on top of the waferndash The word ldquobeamrdquo means that the evaporated atoms

only meet each other on the wafer

Spin Quantum Computing

Qubit information is stored in the spin state of an electron in an artificial atom

AdvantagesLong decoherence time

Future Scalabilty

Artifical atoms are bigger than regular atoms therefore easier to manipulate

Decoherence time ~ 100ns

bull Time before the quantum mechanical system starts acting in a classical way with its complex environment

bull The state of the system has not yet collapsed due to (unwanted) environmental effects

bull Spin - DT are 100 as long as for the Excitonbull Need to SWITCH 104 during DT for reliable error

correction This requirement is met

Artificial Atombull Double Barrier

Heterostructurebull Dot In005Ga095Asbull Source ampDrain GaAsbull 2D Electron Gasbull Confine with gate biasbull D ~ Fermi wavelength

rarr Discrete energy levels

Adding Electrons changing Vgate

bull 2D-Harmonic Oscillator

bull Shell structure as in atoms

bull Magic Numbers 2 6 12

bull To add ldquoevenrdquo electron requires only additional Coulomb energy

Comparison with Hydrogenbull Artificial Atom

Energy levels ~ 1meV

Size ~ 10μm

Weak magnetic fields can affect energy levels

bull Hydrogen

Energy levels ~ 1eV

Size ~ 1Aring

Only strong magnetic fields can perturb energy levels

Factor 1000

Tuning the Quantum Dot

bull Tune so we have one valence electron

bull Initial state can be set by applying homogeneous magnetic field rarr |0gt

bull Low temperature kT lt ΔE (state gap)

bull Now we have defined our single qubit

Energy

position

Gate bias

Spin up - electron

Unoccupied state

Single Qubit manipulation

bull Unitary operations can be made by applying a local magnetic field H

ZE = -μB = g

μB SB

bull MF microscopebull AF microscopebull Sub grid of currentbull Magnetic dotsbull Etc

(Magnetic force microscope tip)

Two Qubit Manipulation

bull Complete set of logic requires a CNOT

bull Dots are placed so close that they overlap and interact

bull Hspin

= J(t)S1S

2

Exchange couplingJ(tEB) = E

triplet -E

singlet

(4th order Harmonic Oscillator)

Ground State Splitting (J = Et ndash E

s)

bull 2 coupled fermions must have an total anti-symmetric wave function

bull Lowest coupled state is the singlet It has a symmetric spatial wave function and an anti symmetric spin (Coulomb dominates)|ψ

sgt ~ (|12gt + |21gt) (|darruarrgt - |uarrdarrgt)

bull The triplet states are (|darrdarrgt)|ψ

tgt ~ (|12gt - |21gt) (|darruarrgt + |uarrdarrgt)

(|uarruarrgt)bull lt1|2gt ne 0 |igt is spatial wf Coulomb dominates

Solving J(B(t)) Exchange Coupling

bull Different solutions Heitler-London Hund-Mulliken Hubbard

bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations

SWAP - gatebull Assume J can be pulsed

J(t) = 0 J0

Formula 1

Formula 2

bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]

XOR ~ CNOT

bull Formula 3

bull Requirements Spin rotations about the z-axis Squareroot of U

swap

Read out Memory

bull Assume dot with an electron with some information stored in spin-state

bull Connect two leads to dot

bull Apply a small bias (ΔV) rarr Current (i)Energy

position

Gate bias

Spin up - electron

Unoccupied state

i

Another Spin up electron enters dot

bull Pauli principle forces electrons with spin up to occupy the higher energy state

bull Negligible chance of tunneling

E

position

Gate bias

Spin up - electron

Higher energy level(forbidden classically)

i=0

Spin down electron enters dot

bull Pauli principle allows the new electron to join the same energy level as the original electron

bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow

E

position

Gate bias

Spin up - electron

Unoccupied state

ine0

Read out Memory

bull We have a way of measuring the spin state of an electron in a quantum dot

bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties

bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique

bull The electron current can be specialized (we can aim its spin to make measurement efficient)

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

The Physical System Excitons Trapped in GaAs Quantum Dots

bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid

bull Often formed when photons excite electrons from the valence band into the conduction band

bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses

bull Decay by radiating photons Decay time ~50ps-1ns

bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt

Initialization

bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to

prevent thermal excitations

bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the

register

Relatively Long Decoherence Times

bull Mechanismsndash Radiative Decay ~10ps-1ns

bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations

bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral

bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps

or less

A Universal set of Quantum Gates

bull Single Qubit Rotations through laser induced Rabi Oscillations

bull CNOT operations through dipole interactions and laser excitation

Single Qubit Gates Rabi Flopping

bull Light-particle interaction is characterized by the product of the dipole moment and the electric field

μbullE(t)= ħR(t)

Where R(t) is the Rabi frequency and the pulse area is given by

Θ(t)=intR(t)dt

and the state at time t is then given by

Cos(Θ2)|0gt+Sin(Θ2)|1gt

Stufler et al

Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses

Cos(Θ2)|0gt+Sin(Θ2)|1gt

|1gt =gt electric charge

=gtPhotocurrent (PC)PC~Sin2(Θ2)

π-pulse corresponds to a population inversion

CNOT Dipole Coupling

Nearest neighbor interactions alter the energy states

Effective energy Ersquoi = Ei + sumjnei ∆Eij nj

Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied

Overcoming Short Interaction Distances

bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs

bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots

bull Optical Cavity coupling and fiber optical interconnects have also been proposed

Read Out of Specified Qubit States

bull Optical readoutExcitons decay spontaneously and the resulting radiation can be

measured

Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD

Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination

In either case repeated measurements have to be made A single shot readout still needs to be developed

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

• Quantum Dots
• Implementing Quantum Computers
• What is a quantum dot
• Gallium Arsenide Quantum Dots
• Energy Band Levels
• Slide 6
• Slide 7
• Tuning Quantum Dots
• Manufacturing methods
• Electron Beam Lithography
• Slide 11
• Molecular Beam Epitaxy
• Spin Quantum Computing
• Decoherence time ~ 100ns
• Artificial Atom
• Adding Electrons changing Vgate
• Comparison with Hydrogen
• Tuning the Quantum Dot
• Single Qubit manipulation
• Two Qubit Manipulation
• Ground State Splitting (J = Et ndash Es)
• Solving J(B(t)) Exchange Coupling
• SWAP - gate
• XOR ~ CNOT
• Read out Memory
• Another Spin up electron enters dot
• Spin down electron enters dot
• Slide 28
• 5 DiVincenzo QC Criteria
• The Physical System Excitons Trapped in GaAs Quantum Dots
• Initialization
• Relatively Long Decoherence Times
• A Universal set of Quantum Gates
• Single Qubit Gates Rabi Flopping
• Stufler et al
• CNOT Dipole Coupling
• Overcoming Short Interaction Distances
• Read Out of Specified Qubit States
• Slide 39

Electron Beam Lithography

bull Electrons are accelerated out of an electron gun and sent through condenser lens optics directly onto a wafer

bull λ = (123 Aring radicV)bull Advantages

ndash generation of micron and submicron resist geometries

ndash greater depth of focus than optical lithography

ndash masks are unnecessaryndash Optical diffraction limit is not a

real concern

Electron Beam Lithography

bull Disadvantage(s)ndash The lithography is serial

(masks arenrsquot used instead the beam itself sweeps across the wafer) =gt Comparatively low throughput ~5 wafers per hour at less than 1 micrometer resolution

ndash The proximity effect Electrons scatter because they are relatively low in mass reducing the resolution

bull Heavy ion lithography has been proposed but still is in development stages

Molecular Beam Epitaxy

bull Molecular beam epitaxy (MBE) is the deposition of one or more pure materials onto a single crystal wafer one layer of atoms at a time in order to form a perfect crystalndash This is done by evaporating each of the elements to

combine then condensing them on top of the waferndash The word ldquobeamrdquo means that the evaporated atoms

only meet each other on the wafer

Spin Quantum Computing

Qubit information is stored in the spin state of an electron in an artificial atom

AdvantagesLong decoherence time

Future Scalabilty

Artifical atoms are bigger than regular atoms therefore easier to manipulate

Decoherence time ~ 100ns

bull Time before the quantum mechanical system starts acting in a classical way with its complex environment

bull The state of the system has not yet collapsed due to (unwanted) environmental effects

bull Spin - DT are 100 as long as for the Excitonbull Need to SWITCH 104 during DT for reliable error

correction This requirement is met

Artificial Atombull Double Barrier

Heterostructurebull Dot In005Ga095Asbull Source ampDrain GaAsbull 2D Electron Gasbull Confine with gate biasbull D ~ Fermi wavelength

rarr Discrete energy levels

Adding Electrons changing Vgate

bull 2D-Harmonic Oscillator

bull Shell structure as in atoms

bull Magic Numbers 2 6 12

bull To add ldquoevenrdquo electron requires only additional Coulomb energy

Comparison with Hydrogenbull Artificial Atom

Energy levels ~ 1meV

Size ~ 10μm

Weak magnetic fields can affect energy levels

bull Hydrogen

Energy levels ~ 1eV

Size ~ 1Aring

Only strong magnetic fields can perturb energy levels

Factor 1000

Tuning the Quantum Dot

bull Tune so we have one valence electron

bull Initial state can be set by applying homogeneous magnetic field rarr |0gt

bull Low temperature kT lt ΔE (state gap)

bull Now we have defined our single qubit

Energy

position

Gate bias

Spin up - electron

Unoccupied state

Single Qubit manipulation

bull Unitary operations can be made by applying a local magnetic field H

ZE = -μB = g

μB SB

bull MF microscopebull AF microscopebull Sub grid of currentbull Magnetic dotsbull Etc

(Magnetic force microscope tip)

Two Qubit Manipulation

bull Complete set of logic requires a CNOT

bull Dots are placed so close that they overlap and interact

bull Hspin

= J(t)S1S

2

Exchange couplingJ(tEB) = E

triplet -E

singlet

(4th order Harmonic Oscillator)

Ground State Splitting (J = Et ndash E

s)

bull 2 coupled fermions must have an total anti-symmetric wave function

bull Lowest coupled state is the singlet It has a symmetric spatial wave function and an anti symmetric spin (Coulomb dominates)|ψ

sgt ~ (|12gt + |21gt) (|darruarrgt - |uarrdarrgt)

bull The triplet states are (|darrdarrgt)|ψ

tgt ~ (|12gt - |21gt) (|darruarrgt + |uarrdarrgt)

(|uarruarrgt)bull lt1|2gt ne 0 |igt is spatial wf Coulomb dominates

Solving J(B(t)) Exchange Coupling

bull Different solutions Heitler-London Hund-Mulliken Hubbard

bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations

SWAP - gatebull Assume J can be pulsed

J(t) = 0 J0

Formula 1

Formula 2

bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]

XOR ~ CNOT

bull Formula 3

bull Requirements Spin rotations about the z-axis Squareroot of U

swap

Read out Memory

bull Assume dot with an electron with some information stored in spin-state

bull Connect two leads to dot

bull Apply a small bias (ΔV) rarr Current (i)Energy

position

Gate bias

Spin up - electron

Unoccupied state

i

Another Spin up electron enters dot

bull Pauli principle forces electrons with spin up to occupy the higher energy state

bull Negligible chance of tunneling

E

position

Gate bias

Spin up - electron

Higher energy level(forbidden classically)

i=0

Spin down electron enters dot

bull Pauli principle allows the new electron to join the same energy level as the original electron

bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow

E

position

Gate bias

Spin up - electron

Unoccupied state

ine0

Read out Memory

bull We have a way of measuring the spin state of an electron in a quantum dot

bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties

bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique

bull The electron current can be specialized (we can aim its spin to make measurement efficient)

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

The Physical System Excitons Trapped in GaAs Quantum Dots

bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid

bull Often formed when photons excite electrons from the valence band into the conduction band

bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses

bull Decay by radiating photons Decay time ~50ps-1ns

bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt

Initialization

bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to

prevent thermal excitations

bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the

register

Relatively Long Decoherence Times

bull Mechanismsndash Radiative Decay ~10ps-1ns

bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations

bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral

bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps

or less

A Universal set of Quantum Gates

bull Single Qubit Rotations through laser induced Rabi Oscillations

bull CNOT operations through dipole interactions and laser excitation

Single Qubit Gates Rabi Flopping

bull Light-particle interaction is characterized by the product of the dipole moment and the electric field

μbullE(t)= ħR(t)

Where R(t) is the Rabi frequency and the pulse area is given by

Θ(t)=intR(t)dt

and the state at time t is then given by

Cos(Θ2)|0gt+Sin(Θ2)|1gt

Stufler et al

Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses

Cos(Θ2)|0gt+Sin(Θ2)|1gt

|1gt =gt electric charge

=gtPhotocurrent (PC)PC~Sin2(Θ2)

π-pulse corresponds to a population inversion

CNOT Dipole Coupling

Nearest neighbor interactions alter the energy states

Effective energy Ersquoi = Ei + sumjnei ∆Eij nj

Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied

Overcoming Short Interaction Distances

bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs

bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots

bull Optical Cavity coupling and fiber optical interconnects have also been proposed

Read Out of Specified Qubit States

bull Optical readoutExcitons decay spontaneously and the resulting radiation can be

measured

Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD

Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination

In either case repeated measurements have to be made A single shot readout still needs to be developed

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

• Quantum Dots
• Implementing Quantum Computers
• What is a quantum dot
• Gallium Arsenide Quantum Dots
• Energy Band Levels
• Slide 6
• Slide 7
• Tuning Quantum Dots
• Manufacturing methods
• Electron Beam Lithography
• Slide 11
• Molecular Beam Epitaxy
• Spin Quantum Computing
• Decoherence time ~ 100ns
• Artificial Atom
• Adding Electrons changing Vgate
• Comparison with Hydrogen
• Tuning the Quantum Dot
• Single Qubit manipulation
• Two Qubit Manipulation
• Ground State Splitting (J = Et ndash Es)
• Solving J(B(t)) Exchange Coupling
• SWAP - gate
• XOR ~ CNOT
• Read out Memory
• Another Spin up electron enters dot
• Spin down electron enters dot
• Slide 28
• 5 DiVincenzo QC Criteria
• The Physical System Excitons Trapped in GaAs Quantum Dots
• Initialization
• Relatively Long Decoherence Times
• A Universal set of Quantum Gates
• Single Qubit Gates Rabi Flopping
• Stufler et al
• CNOT Dipole Coupling
• Overcoming Short Interaction Distances
• Read Out of Specified Qubit States
• Slide 39

Electron Beam Lithography

bull Disadvantage(s)ndash The lithography is serial

(masks arenrsquot used instead the beam itself sweeps across the wafer) =gt Comparatively low throughput ~5 wafers per hour at less than 1 micrometer resolution

ndash The proximity effect Electrons scatter because they are relatively low in mass reducing the resolution

bull Heavy ion lithography has been proposed but still is in development stages

Molecular Beam Epitaxy

bull Molecular beam epitaxy (MBE) is the deposition of one or more pure materials onto a single crystal wafer one layer of atoms at a time in order to form a perfect crystalndash This is done by evaporating each of the elements to

combine then condensing them on top of the waferndash The word ldquobeamrdquo means that the evaporated atoms

only meet each other on the wafer

Spin Quantum Computing

Qubit information is stored in the spin state of an electron in an artificial atom

AdvantagesLong decoherence time

Future Scalabilty

Artifical atoms are bigger than regular atoms therefore easier to manipulate

Decoherence time ~ 100ns

bull Time before the quantum mechanical system starts acting in a classical way with its complex environment

bull The state of the system has not yet collapsed due to (unwanted) environmental effects

bull Spin - DT are 100 as long as for the Excitonbull Need to SWITCH 104 during DT for reliable error

correction This requirement is met

Artificial Atombull Double Barrier

Heterostructurebull Dot In005Ga095Asbull Source ampDrain GaAsbull 2D Electron Gasbull Confine with gate biasbull D ~ Fermi wavelength

rarr Discrete energy levels

Adding Electrons changing Vgate

bull 2D-Harmonic Oscillator

bull Shell structure as in atoms

bull Magic Numbers 2 6 12

bull To add ldquoevenrdquo electron requires only additional Coulomb energy

Comparison with Hydrogenbull Artificial Atom

Energy levels ~ 1meV

Size ~ 10μm

Weak magnetic fields can affect energy levels

bull Hydrogen

Energy levels ~ 1eV

Size ~ 1Aring

Only strong magnetic fields can perturb energy levels

Factor 1000

Tuning the Quantum Dot

bull Tune so we have one valence electron

bull Initial state can be set by applying homogeneous magnetic field rarr |0gt

bull Low temperature kT lt ΔE (state gap)

bull Now we have defined our single qubit

Energy

position

Gate bias

Spin up - electron

Unoccupied state

Single Qubit manipulation

bull Unitary operations can be made by applying a local magnetic field H

ZE = -μB = g

μB SB

bull MF microscopebull AF microscopebull Sub grid of currentbull Magnetic dotsbull Etc

(Magnetic force microscope tip)

Two Qubit Manipulation

bull Complete set of logic requires a CNOT

bull Dots are placed so close that they overlap and interact

bull Hspin

= J(t)S1S

2

Exchange couplingJ(tEB) = E

triplet -E

singlet

(4th order Harmonic Oscillator)

Ground State Splitting (J = Et ndash E

s)

bull 2 coupled fermions must have an total anti-symmetric wave function

bull Lowest coupled state is the singlet It has a symmetric spatial wave function and an anti symmetric spin (Coulomb dominates)|ψ

sgt ~ (|12gt + |21gt) (|darruarrgt - |uarrdarrgt)

bull The triplet states are (|darrdarrgt)|ψ

tgt ~ (|12gt - |21gt) (|darruarrgt + |uarrdarrgt)

(|uarruarrgt)bull lt1|2gt ne 0 |igt is spatial wf Coulomb dominates

Solving J(B(t)) Exchange Coupling

bull Different solutions Heitler-London Hund-Mulliken Hubbard

bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations

SWAP - gatebull Assume J can be pulsed

J(t) = 0 J0

Formula 1

Formula 2

bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]

XOR ~ CNOT

bull Formula 3

bull Requirements Spin rotations about the z-axis Squareroot of U

swap

Read out Memory

bull Assume dot with an electron with some information stored in spin-state

bull Connect two leads to dot

bull Apply a small bias (ΔV) rarr Current (i)Energy

position

Gate bias

Spin up - electron

Unoccupied state

i

Another Spin up electron enters dot

bull Pauli principle forces electrons with spin up to occupy the higher energy state

bull Negligible chance of tunneling

E

position

Gate bias

Spin up - electron

Higher energy level(forbidden classically)

i=0

Spin down electron enters dot

bull Pauli principle allows the new electron to join the same energy level as the original electron

bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow

E

position

Gate bias

Spin up - electron

Unoccupied state

ine0

Read out Memory

bull We have a way of measuring the spin state of an electron in a quantum dot

bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties

bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique

bull The electron current can be specialized (we can aim its spin to make measurement efficient)

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

The Physical System Excitons Trapped in GaAs Quantum Dots

bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid

bull Often formed when photons excite electrons from the valence band into the conduction band

bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses

bull Decay by radiating photons Decay time ~50ps-1ns

bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt

Initialization

bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to

prevent thermal excitations

bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the

register

Relatively Long Decoherence Times

bull Mechanismsndash Radiative Decay ~10ps-1ns

bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations

bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral

bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps

or less

A Universal set of Quantum Gates

bull Single Qubit Rotations through laser induced Rabi Oscillations

bull CNOT operations through dipole interactions and laser excitation

Single Qubit Gates Rabi Flopping

bull Light-particle interaction is characterized by the product of the dipole moment and the electric field

μbullE(t)= ħR(t)

Where R(t) is the Rabi frequency and the pulse area is given by

Θ(t)=intR(t)dt

and the state at time t is then given by

Cos(Θ2)|0gt+Sin(Θ2)|1gt

Stufler et al

Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses

Cos(Θ2)|0gt+Sin(Θ2)|1gt

|1gt =gt electric charge

=gtPhotocurrent (PC)PC~Sin2(Θ2)

π-pulse corresponds to a population inversion

CNOT Dipole Coupling

Nearest neighbor interactions alter the energy states

Effective energy Ersquoi = Ei + sumjnei ∆Eij nj

Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied

Overcoming Short Interaction Distances

bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs

bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots

bull Optical Cavity coupling and fiber optical interconnects have also been proposed

Read Out of Specified Qubit States

bull Optical readoutExcitons decay spontaneously and the resulting radiation can be

measured

Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD

Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination

In either case repeated measurements have to be made A single shot readout still needs to be developed

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

• Quantum Dots
• Implementing Quantum Computers
• What is a quantum dot
• Gallium Arsenide Quantum Dots
• Energy Band Levels
• Slide 6
• Slide 7
• Tuning Quantum Dots
• Manufacturing methods
• Electron Beam Lithography
• Slide 11
• Molecular Beam Epitaxy
• Spin Quantum Computing
• Decoherence time ~ 100ns
• Artificial Atom
• Adding Electrons changing Vgate
• Comparison with Hydrogen
• Tuning the Quantum Dot
• Single Qubit manipulation
• Two Qubit Manipulation
• Ground State Splitting (J = Et ndash Es)
• Solving J(B(t)) Exchange Coupling
• SWAP - gate
• XOR ~ CNOT
• Read out Memory
• Another Spin up electron enters dot
• Spin down electron enters dot
• Slide 28
• 5 DiVincenzo QC Criteria
• The Physical System Excitons Trapped in GaAs Quantum Dots
• Initialization
• Relatively Long Decoherence Times
• A Universal set of Quantum Gates
• Single Qubit Gates Rabi Flopping
• Stufler et al
• CNOT Dipole Coupling
• Overcoming Short Interaction Distances
• Read Out of Specified Qubit States
• Slide 39

Molecular Beam Epitaxy

bull Molecular beam epitaxy (MBE) is the deposition of one or more pure materials onto a single crystal wafer one layer of atoms at a time in order to form a perfect crystalndash This is done by evaporating each of the elements to

combine then condensing them on top of the waferndash The word ldquobeamrdquo means that the evaporated atoms

only meet each other on the wafer

Spin Quantum Computing

Qubit information is stored in the spin state of an electron in an artificial atom

AdvantagesLong decoherence time

Future Scalabilty

Artifical atoms are bigger than regular atoms therefore easier to manipulate

Decoherence time ~ 100ns

bull Time before the quantum mechanical system starts acting in a classical way with its complex environment

bull The state of the system has not yet collapsed due to (unwanted) environmental effects

bull Spin - DT are 100 as long as for the Excitonbull Need to SWITCH 104 during DT for reliable error

correction This requirement is met

Artificial Atombull Double Barrier

Heterostructurebull Dot In005Ga095Asbull Source ampDrain GaAsbull 2D Electron Gasbull Confine with gate biasbull D ~ Fermi wavelength

rarr Discrete energy levels

Adding Electrons changing Vgate

bull 2D-Harmonic Oscillator

bull Shell structure as in atoms

bull Magic Numbers 2 6 12

bull To add ldquoevenrdquo electron requires only additional Coulomb energy

Comparison with Hydrogenbull Artificial Atom

Energy levels ~ 1meV

Size ~ 10μm

Weak magnetic fields can affect energy levels

bull Hydrogen

Energy levels ~ 1eV

Size ~ 1Aring

Only strong magnetic fields can perturb energy levels

Factor 1000

Tuning the Quantum Dot

bull Tune so we have one valence electron

bull Initial state can be set by applying homogeneous magnetic field rarr |0gt

bull Low temperature kT lt ΔE (state gap)

bull Now we have defined our single qubit

Energy

position

Gate bias

Spin up - electron

Unoccupied state

Single Qubit manipulation

bull Unitary operations can be made by applying a local magnetic field H

ZE = -μB = g

μB SB

bull MF microscopebull AF microscopebull Sub grid of currentbull Magnetic dotsbull Etc

(Magnetic force microscope tip)

Two Qubit Manipulation

bull Complete set of logic requires a CNOT

bull Dots are placed so close that they overlap and interact

bull Hspin

= J(t)S1S

2

Exchange couplingJ(tEB) = E

triplet -E

singlet

(4th order Harmonic Oscillator)

Ground State Splitting (J = Et ndash E

s)

bull 2 coupled fermions must have an total anti-symmetric wave function

bull Lowest coupled state is the singlet It has a symmetric spatial wave function and an anti symmetric spin (Coulomb dominates)|ψ

sgt ~ (|12gt + |21gt) (|darruarrgt - |uarrdarrgt)

bull The triplet states are (|darrdarrgt)|ψ

tgt ~ (|12gt - |21gt) (|darruarrgt + |uarrdarrgt)

(|uarruarrgt)bull lt1|2gt ne 0 |igt is spatial wf Coulomb dominates

Solving J(B(t)) Exchange Coupling

bull Different solutions Heitler-London Hund-Mulliken Hubbard

bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations

SWAP - gatebull Assume J can be pulsed

J(t) = 0 J0

Formula 1

Formula 2

bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]

XOR ~ CNOT

bull Formula 3

bull Requirements Spin rotations about the z-axis Squareroot of U

swap

Read out Memory

bull Assume dot with an electron with some information stored in spin-state

bull Connect two leads to dot

bull Apply a small bias (ΔV) rarr Current (i)Energy

position

Gate bias

Spin up - electron

Unoccupied state

i

Another Spin up electron enters dot

bull Pauli principle forces electrons with spin up to occupy the higher energy state

bull Negligible chance of tunneling

E

position

Gate bias

Spin up - electron

Higher energy level(forbidden classically)

i=0

Spin down electron enters dot

bull Pauli principle allows the new electron to join the same energy level as the original electron

bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow

E

position

Gate bias

Spin up - electron

Unoccupied state

ine0

Read out Memory

bull We have a way of measuring the spin state of an electron in a quantum dot

bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties

bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique

bull The electron current can be specialized (we can aim its spin to make measurement efficient)

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

The Physical System Excitons Trapped in GaAs Quantum Dots

bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid

bull Often formed when photons excite electrons from the valence band into the conduction band

bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses

bull Decay by radiating photons Decay time ~50ps-1ns

bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt

Initialization

bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to

prevent thermal excitations

bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the

register

Relatively Long Decoherence Times

bull Mechanismsndash Radiative Decay ~10ps-1ns

bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations

bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral

bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps

or less

A Universal set of Quantum Gates

bull Single Qubit Rotations through laser induced Rabi Oscillations

bull CNOT operations through dipole interactions and laser excitation

Single Qubit Gates Rabi Flopping

bull Light-particle interaction is characterized by the product of the dipole moment and the electric field

μbullE(t)= ħR(t)

Where R(t) is the Rabi frequency and the pulse area is given by

Θ(t)=intR(t)dt

and the state at time t is then given by

Cos(Θ2)|0gt+Sin(Θ2)|1gt

Stufler et al

Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses

Cos(Θ2)|0gt+Sin(Θ2)|1gt

|1gt =gt electric charge

=gtPhotocurrent (PC)PC~Sin2(Θ2)

π-pulse corresponds to a population inversion

CNOT Dipole Coupling

Nearest neighbor interactions alter the energy states

Effective energy Ersquoi = Ei + sumjnei ∆Eij nj

Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied

Overcoming Short Interaction Distances

bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs

bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots

bull Optical Cavity coupling and fiber optical interconnects have also been proposed

Read Out of Specified Qubit States

bull Optical readoutExcitons decay spontaneously and the resulting radiation can be

measured

Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD

Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination

In either case repeated measurements have to be made A single shot readout still needs to be developed

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

• Quantum Dots
• Implementing Quantum Computers
• What is a quantum dot
• Gallium Arsenide Quantum Dots
• Energy Band Levels
• Slide 6
• Slide 7
• Tuning Quantum Dots
• Manufacturing methods
• Electron Beam Lithography
• Slide 11
• Molecular Beam Epitaxy
• Spin Quantum Computing
• Decoherence time ~ 100ns
• Artificial Atom
• Adding Electrons changing Vgate
• Comparison with Hydrogen
• Tuning the Quantum Dot
• Single Qubit manipulation
• Two Qubit Manipulation
• Ground State Splitting (J = Et ndash Es)
• Solving J(B(t)) Exchange Coupling
• SWAP - gate
• XOR ~ CNOT
• Read out Memory
• Another Spin up electron enters dot
• Spin down electron enters dot
• Slide 28
• 5 DiVincenzo QC Criteria
• The Physical System Excitons Trapped in GaAs Quantum Dots
• Initialization
• Relatively Long Decoherence Times
• A Universal set of Quantum Gates
• Single Qubit Gates Rabi Flopping
• Stufler et al
• CNOT Dipole Coupling
• Overcoming Short Interaction Distances
• Read Out of Specified Qubit States
• Slide 39

Spin Quantum Computing

Qubit information is stored in the spin state of an electron in an artificial atom

AdvantagesLong decoherence time

Future Scalabilty

Artifical atoms are bigger than regular atoms therefore easier to manipulate

Decoherence time ~ 100ns

bull Time before the quantum mechanical system starts acting in a classical way with its complex environment

bull The state of the system has not yet collapsed due to (unwanted) environmental effects

bull Spin - DT are 100 as long as for the Excitonbull Need to SWITCH 104 during DT for reliable error

correction This requirement is met

Artificial Atombull Double Barrier

Heterostructurebull Dot In005Ga095Asbull Source ampDrain GaAsbull 2D Electron Gasbull Confine with gate biasbull D ~ Fermi wavelength

rarr Discrete energy levels

Adding Electrons changing Vgate

bull 2D-Harmonic Oscillator

bull Shell structure as in atoms

bull Magic Numbers 2 6 12

bull To add ldquoevenrdquo electron requires only additional Coulomb energy

Comparison with Hydrogenbull Artificial Atom

Energy levels ~ 1meV

Size ~ 10μm

Weak magnetic fields can affect energy levels

bull Hydrogen

Energy levels ~ 1eV

Size ~ 1Aring

Only strong magnetic fields can perturb energy levels

Factor 1000

Tuning the Quantum Dot

bull Tune so we have one valence electron

bull Initial state can be set by applying homogeneous magnetic field rarr |0gt

bull Low temperature kT lt ΔE (state gap)

bull Now we have defined our single qubit

Energy

position

Gate bias

Spin up - electron

Unoccupied state

Single Qubit manipulation

bull Unitary operations can be made by applying a local magnetic field H

ZE = -μB = g

μB SB

bull MF microscopebull AF microscopebull Sub grid of currentbull Magnetic dotsbull Etc

(Magnetic force microscope tip)

Two Qubit Manipulation

bull Complete set of logic requires a CNOT

bull Dots are placed so close that they overlap and interact

bull Hspin

= J(t)S1S

2

Exchange couplingJ(tEB) = E

triplet -E

singlet

(4th order Harmonic Oscillator)

Ground State Splitting (J = Et ndash E

s)

bull 2 coupled fermions must have an total anti-symmetric wave function

bull Lowest coupled state is the singlet It has a symmetric spatial wave function and an anti symmetric spin (Coulomb dominates)|ψ

sgt ~ (|12gt + |21gt) (|darruarrgt - |uarrdarrgt)

bull The triplet states are (|darrdarrgt)|ψ

tgt ~ (|12gt - |21gt) (|darruarrgt + |uarrdarrgt)

(|uarruarrgt)bull lt1|2gt ne 0 |igt is spatial wf Coulomb dominates

Solving J(B(t)) Exchange Coupling

bull Different solutions Heitler-London Hund-Mulliken Hubbard

bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations

SWAP - gatebull Assume J can be pulsed

J(t) = 0 J0

Formula 1

Formula 2

bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]

XOR ~ CNOT

bull Formula 3

bull Requirements Spin rotations about the z-axis Squareroot of U

swap

Read out Memory

bull Assume dot with an electron with some information stored in spin-state

bull Connect two leads to dot

bull Apply a small bias (ΔV) rarr Current (i)Energy

position

Gate bias

Spin up - electron

Unoccupied state

i

Another Spin up electron enters dot

bull Pauli principle forces electrons with spin up to occupy the higher energy state

bull Negligible chance of tunneling

E

position

Gate bias

Spin up - electron

Higher energy level(forbidden classically)

i=0

Spin down electron enters dot

bull Pauli principle allows the new electron to join the same energy level as the original electron

bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow

E

position

Gate bias

Spin up - electron

Unoccupied state

ine0

Read out Memory

bull We have a way of measuring the spin state of an electron in a quantum dot

bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties

bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique

bull The electron current can be specialized (we can aim its spin to make measurement efficient)

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

The Physical System Excitons Trapped in GaAs Quantum Dots

bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid

bull Often formed when photons excite electrons from the valence band into the conduction band

bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses

bull Decay by radiating photons Decay time ~50ps-1ns

bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt

Initialization

bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to

prevent thermal excitations

bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the

register

Relatively Long Decoherence Times

bull Mechanismsndash Radiative Decay ~10ps-1ns

bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations

bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral

bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps

or less

A Universal set of Quantum Gates

bull Single Qubit Rotations through laser induced Rabi Oscillations

bull CNOT operations through dipole interactions and laser excitation

Single Qubit Gates Rabi Flopping

bull Light-particle interaction is characterized by the product of the dipole moment and the electric field

μbullE(t)= ħR(t)

Where R(t) is the Rabi frequency and the pulse area is given by

Θ(t)=intR(t)dt

and the state at time t is then given by

Cos(Θ2)|0gt+Sin(Θ2)|1gt

Stufler et al

Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses

Cos(Θ2)|0gt+Sin(Θ2)|1gt

|1gt =gt electric charge

=gtPhotocurrent (PC)PC~Sin2(Θ2)

π-pulse corresponds to a population inversion

CNOT Dipole Coupling

Nearest neighbor interactions alter the energy states

Effective energy Ersquoi = Ei + sumjnei ∆Eij nj

Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied

Overcoming Short Interaction Distances

bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs

bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots

bull Optical Cavity coupling and fiber optical interconnects have also been proposed

Read Out of Specified Qubit States

bull Optical readoutExcitons decay spontaneously and the resulting radiation can be

measured

Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD

Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination

In either case repeated measurements have to be made A single shot readout still needs to be developed

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

• Quantum Dots
• Implementing Quantum Computers
• What is a quantum dot
• Gallium Arsenide Quantum Dots
• Energy Band Levels
• Slide 6
• Slide 7
• Tuning Quantum Dots
• Manufacturing methods
• Electron Beam Lithography
• Slide 11
• Molecular Beam Epitaxy
• Spin Quantum Computing
• Decoherence time ~ 100ns
• Artificial Atom
• Adding Electrons changing Vgate
• Comparison with Hydrogen
• Tuning the Quantum Dot
• Single Qubit manipulation
• Two Qubit Manipulation
• Ground State Splitting (J = Et ndash Es)
• Solving J(B(t)) Exchange Coupling
• SWAP - gate
• XOR ~ CNOT
• Read out Memory
• Another Spin up electron enters dot
• Spin down electron enters dot
• Slide 28
• 5 DiVincenzo QC Criteria
• The Physical System Excitons Trapped in GaAs Quantum Dots
• Initialization
• Relatively Long Decoherence Times
• A Universal set of Quantum Gates
• Single Qubit Gates Rabi Flopping
• Stufler et al
• CNOT Dipole Coupling
• Overcoming Short Interaction Distances
• Read Out of Specified Qubit States
• Slide 39

Decoherence time ~ 100ns

bull Time before the quantum mechanical system starts acting in a classical way with its complex environment

bull The state of the system has not yet collapsed due to (unwanted) environmental effects

bull Spin - DT are 100 as long as for the Excitonbull Need to SWITCH 104 during DT for reliable error

correction This requirement is met

Artificial Atombull Double Barrier

Heterostructurebull Dot In005Ga095Asbull Source ampDrain GaAsbull 2D Electron Gasbull Confine with gate biasbull D ~ Fermi wavelength

rarr Discrete energy levels

Adding Electrons changing Vgate

bull 2D-Harmonic Oscillator

bull Shell structure as in atoms

bull Magic Numbers 2 6 12

bull To add ldquoevenrdquo electron requires only additional Coulomb energy

Comparison with Hydrogenbull Artificial Atom

Energy levels ~ 1meV

Size ~ 10μm

Weak magnetic fields can affect energy levels

bull Hydrogen

Energy levels ~ 1eV

Size ~ 1Aring

Only strong magnetic fields can perturb energy levels

Factor 1000

Tuning the Quantum Dot

bull Tune so we have one valence electron

bull Initial state can be set by applying homogeneous magnetic field rarr |0gt

bull Low temperature kT lt ΔE (state gap)

bull Now we have defined our single qubit

Energy

position

Gate bias

Spin up - electron

Unoccupied state

Single Qubit manipulation

bull Unitary operations can be made by applying a local magnetic field H

ZE = -μB = g

μB SB

bull MF microscopebull AF microscopebull Sub grid of currentbull Magnetic dotsbull Etc

(Magnetic force microscope tip)

Two Qubit Manipulation

bull Complete set of logic requires a CNOT

bull Dots are placed so close that they overlap and interact

bull Hspin

= J(t)S1S

2

Exchange couplingJ(tEB) = E

triplet -E

singlet

(4th order Harmonic Oscillator)

Ground State Splitting (J = Et ndash E

s)

bull 2 coupled fermions must have an total anti-symmetric wave function

bull Lowest coupled state is the singlet It has a symmetric spatial wave function and an anti symmetric spin (Coulomb dominates)|ψ

sgt ~ (|12gt + |21gt) (|darruarrgt - |uarrdarrgt)

bull The triplet states are (|darrdarrgt)|ψ

tgt ~ (|12gt - |21gt) (|darruarrgt + |uarrdarrgt)

(|uarruarrgt)bull lt1|2gt ne 0 |igt is spatial wf Coulomb dominates

Solving J(B(t)) Exchange Coupling

bull Different solutions Heitler-London Hund-Mulliken Hubbard

bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations

SWAP - gatebull Assume J can be pulsed

J(t) = 0 J0

Formula 1

Formula 2

bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]

XOR ~ CNOT

bull Formula 3

bull Requirements Spin rotations about the z-axis Squareroot of U

swap

Read out Memory

bull Assume dot with an electron with some information stored in spin-state

bull Connect two leads to dot

bull Apply a small bias (ΔV) rarr Current (i)Energy

position

Gate bias

Spin up - electron

Unoccupied state

i

Another Spin up electron enters dot

bull Pauli principle forces electrons with spin up to occupy the higher energy state

bull Negligible chance of tunneling

E

position

Gate bias

Spin up - electron

Higher energy level(forbidden classically)

i=0

Spin down electron enters dot

bull Pauli principle allows the new electron to join the same energy level as the original electron

bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow

E

position

Gate bias

Spin up - electron

Unoccupied state

ine0

Read out Memory

bull We have a way of measuring the spin state of an electron in a quantum dot

bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties

bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique

bull The electron current can be specialized (we can aim its spin to make measurement efficient)

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

The Physical System Excitons Trapped in GaAs Quantum Dots

bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid

bull Often formed when photons excite electrons from the valence band into the conduction band

bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses

bull Decay by radiating photons Decay time ~50ps-1ns

bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt

Initialization

bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to

prevent thermal excitations

bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the

register

Relatively Long Decoherence Times

bull Mechanismsndash Radiative Decay ~10ps-1ns

bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations

bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral

bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps

or less

A Universal set of Quantum Gates

bull Single Qubit Rotations through laser induced Rabi Oscillations

bull CNOT operations through dipole interactions and laser excitation

Single Qubit Gates Rabi Flopping

bull Light-particle interaction is characterized by the product of the dipole moment and the electric field

μbullE(t)= ħR(t)

Where R(t) is the Rabi frequency and the pulse area is given by

Θ(t)=intR(t)dt

and the state at time t is then given by

Cos(Θ2)|0gt+Sin(Θ2)|1gt

Stufler et al

Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses

Cos(Θ2)|0gt+Sin(Θ2)|1gt

|1gt =gt electric charge

=gtPhotocurrent (PC)PC~Sin2(Θ2)

π-pulse corresponds to a population inversion

CNOT Dipole Coupling

Nearest neighbor interactions alter the energy states

Effective energy Ersquoi = Ei + sumjnei ∆Eij nj

Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied

Overcoming Short Interaction Distances

bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs

bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots

bull Optical Cavity coupling and fiber optical interconnects have also been proposed

Read Out of Specified Qubit States

bull Optical readoutExcitons decay spontaneously and the resulting radiation can be

measured

Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD

Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination

In either case repeated measurements have to be made A single shot readout still needs to be developed

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

• Quantum Dots
• Implementing Quantum Computers
• What is a quantum dot
• Gallium Arsenide Quantum Dots
• Energy Band Levels
• Slide 6
• Slide 7
• Tuning Quantum Dots
• Manufacturing methods
• Electron Beam Lithography
• Slide 11
• Molecular Beam Epitaxy
• Spin Quantum Computing
• Decoherence time ~ 100ns
• Artificial Atom
• Adding Electrons changing Vgate
• Comparison with Hydrogen
• Tuning the Quantum Dot
• Single Qubit manipulation
• Two Qubit Manipulation
• Ground State Splitting (J = Et ndash Es)
• Solving J(B(t)) Exchange Coupling
• SWAP - gate
• XOR ~ CNOT
• Read out Memory
• Another Spin up electron enters dot
• Spin down electron enters dot
• Slide 28
• 5 DiVincenzo QC Criteria
• The Physical System Excitons Trapped in GaAs Quantum Dots
• Initialization
• Relatively Long Decoherence Times
• A Universal set of Quantum Gates
• Single Qubit Gates Rabi Flopping
• Stufler et al
• CNOT Dipole Coupling
• Overcoming Short Interaction Distances
• Read Out of Specified Qubit States
• Slide 39

Artificial Atombull Double Barrier

Heterostructurebull Dot In005Ga095Asbull Source ampDrain GaAsbull 2D Electron Gasbull Confine with gate biasbull D ~ Fermi wavelength

rarr Discrete energy levels

Adding Electrons changing Vgate

bull 2D-Harmonic Oscillator

bull Shell structure as in atoms

bull Magic Numbers 2 6 12

bull To add ldquoevenrdquo electron requires only additional Coulomb energy

Comparison with Hydrogenbull Artificial Atom

Energy levels ~ 1meV

Size ~ 10μm

Weak magnetic fields can affect energy levels

bull Hydrogen

Energy levels ~ 1eV

Size ~ 1Aring

Only strong magnetic fields can perturb energy levels

Factor 1000

Tuning the Quantum Dot

bull Tune so we have one valence electron

bull Initial state can be set by applying homogeneous magnetic field rarr |0gt

bull Low temperature kT lt ΔE (state gap)

bull Now we have defined our single qubit

Energy

position

Gate bias

Spin up - electron

Unoccupied state

Single Qubit manipulation

bull Unitary operations can be made by applying a local magnetic field H

ZE = -μB = g

μB SB

bull MF microscopebull AF microscopebull Sub grid of currentbull Magnetic dotsbull Etc

(Magnetic force microscope tip)

Two Qubit Manipulation

bull Complete set of logic requires a CNOT

bull Dots are placed so close that they overlap and interact

bull Hspin

= J(t)S1S

2

Exchange couplingJ(tEB) = E

triplet -E

singlet

(4th order Harmonic Oscillator)

Ground State Splitting (J = Et ndash E

s)

bull 2 coupled fermions must have an total anti-symmetric wave function

bull Lowest coupled state is the singlet It has a symmetric spatial wave function and an anti symmetric spin (Coulomb dominates)|ψ

sgt ~ (|12gt + |21gt) (|darruarrgt - |uarrdarrgt)

bull The triplet states are (|darrdarrgt)|ψ

tgt ~ (|12gt - |21gt) (|darruarrgt + |uarrdarrgt)

(|uarruarrgt)bull lt1|2gt ne 0 |igt is spatial wf Coulomb dominates

Solving J(B(t)) Exchange Coupling

bull Different solutions Heitler-London Hund-Mulliken Hubbard

bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations

SWAP - gatebull Assume J can be pulsed

J(t) = 0 J0

Formula 1

Formula 2

bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]

XOR ~ CNOT

bull Formula 3

bull Requirements Spin rotations about the z-axis Squareroot of U

swap

Read out Memory

bull Assume dot with an electron with some information stored in spin-state

bull Connect two leads to dot

bull Apply a small bias (ΔV) rarr Current (i)Energy

position

Gate bias

Spin up - electron

Unoccupied state

i

Another Spin up electron enters dot

bull Pauli principle forces electrons with spin up to occupy the higher energy state

bull Negligible chance of tunneling

E

position

Gate bias

Spin up - electron

Higher energy level(forbidden classically)

i=0

Spin down electron enters dot

bull Pauli principle allows the new electron to join the same energy level as the original electron

bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow

E

position

Gate bias

Spin up - electron

Unoccupied state

ine0

Read out Memory

bull We have a way of measuring the spin state of an electron in a quantum dot

bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties

bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique

bull The electron current can be specialized (we can aim its spin to make measurement efficient)

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

The Physical System Excitons Trapped in GaAs Quantum Dots

bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid

bull Often formed when photons excite electrons from the valence band into the conduction band

bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses

bull Decay by radiating photons Decay time ~50ps-1ns

bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt

Initialization

bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to

prevent thermal excitations

bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the

register

Relatively Long Decoherence Times

bull Mechanismsndash Radiative Decay ~10ps-1ns

bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations

bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral

bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps

or less

A Universal set of Quantum Gates

bull Single Qubit Rotations through laser induced Rabi Oscillations

bull CNOT operations through dipole interactions and laser excitation

Single Qubit Gates Rabi Flopping

bull Light-particle interaction is characterized by the product of the dipole moment and the electric field

μbullE(t)= ħR(t)

Where R(t) is the Rabi frequency and the pulse area is given by

Θ(t)=intR(t)dt

and the state at time t is then given by

Cos(Θ2)|0gt+Sin(Θ2)|1gt

Stufler et al

Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses

Cos(Θ2)|0gt+Sin(Θ2)|1gt

|1gt =gt electric charge

=gtPhotocurrent (PC)PC~Sin2(Θ2)

π-pulse corresponds to a population inversion

CNOT Dipole Coupling

Nearest neighbor interactions alter the energy states

Effective energy Ersquoi = Ei + sumjnei ∆Eij nj

Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied

Overcoming Short Interaction Distances

bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs

bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots

bull Optical Cavity coupling and fiber optical interconnects have also been proposed

Read Out of Specified Qubit States

bull Optical readoutExcitons decay spontaneously and the resulting radiation can be

measured

Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD

Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination

In either case repeated measurements have to be made A single shot readout still needs to be developed

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

• Quantum Dots
• Implementing Quantum Computers
• What is a quantum dot
• Gallium Arsenide Quantum Dots
• Energy Band Levels
• Slide 6
• Slide 7
• Tuning Quantum Dots
• Manufacturing methods
• Electron Beam Lithography
• Slide 11
• Molecular Beam Epitaxy
• Spin Quantum Computing
• Decoherence time ~ 100ns
• Artificial Atom
• Adding Electrons changing Vgate
• Comparison with Hydrogen
• Tuning the Quantum Dot
• Single Qubit manipulation
• Two Qubit Manipulation
• Ground State Splitting (J = Et ndash Es)
• Solving J(B(t)) Exchange Coupling
• SWAP - gate
• XOR ~ CNOT
• Read out Memory
• Another Spin up electron enters dot
• Spin down electron enters dot
• Slide 28
• 5 DiVincenzo QC Criteria
• The Physical System Excitons Trapped in GaAs Quantum Dots
• Initialization
• Relatively Long Decoherence Times
• A Universal set of Quantum Gates
• Single Qubit Gates Rabi Flopping
• Stufler et al
• CNOT Dipole Coupling
• Overcoming Short Interaction Distances
• Read Out of Specified Qubit States
• Slide 39

Adding Electrons changing Vgate

bull 2D-Harmonic Oscillator

bull Shell structure as in atoms

bull Magic Numbers 2 6 12

bull To add ldquoevenrdquo electron requires only additional Coulomb energy

Comparison with Hydrogenbull Artificial Atom

Energy levels ~ 1meV

Size ~ 10μm

Weak magnetic fields can affect energy levels

bull Hydrogen

Energy levels ~ 1eV

Size ~ 1Aring

Only strong magnetic fields can perturb energy levels

Factor 1000

Tuning the Quantum Dot

bull Tune so we have one valence electron

bull Initial state can be set by applying homogeneous magnetic field rarr |0gt

bull Low temperature kT lt ΔE (state gap)

bull Now we have defined our single qubit

Energy

position

Gate bias

Spin up - electron

Unoccupied state

Single Qubit manipulation

bull Unitary operations can be made by applying a local magnetic field H

ZE = -μB = g

μB SB

bull MF microscopebull AF microscopebull Sub grid of currentbull Magnetic dotsbull Etc

(Magnetic force microscope tip)

Two Qubit Manipulation

bull Complete set of logic requires a CNOT

bull Dots are placed so close that they overlap and interact

bull Hspin

= J(t)S1S

2

Exchange couplingJ(tEB) = E

triplet -E

singlet

(4th order Harmonic Oscillator)

Ground State Splitting (J = Et ndash E

s)

bull 2 coupled fermions must have an total anti-symmetric wave function

bull Lowest coupled state is the singlet It has a symmetric spatial wave function and an anti symmetric spin (Coulomb dominates)|ψ

sgt ~ (|12gt + |21gt) (|darruarrgt - |uarrdarrgt)

bull The triplet states are (|darrdarrgt)|ψ

tgt ~ (|12gt - |21gt) (|darruarrgt + |uarrdarrgt)

(|uarruarrgt)bull lt1|2gt ne 0 |igt is spatial wf Coulomb dominates

Solving J(B(t)) Exchange Coupling

bull Different solutions Heitler-London Hund-Mulliken Hubbard

bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations

SWAP - gatebull Assume J can be pulsed

J(t) = 0 J0

Formula 1

Formula 2

bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]

XOR ~ CNOT

bull Formula 3

bull Requirements Spin rotations about the z-axis Squareroot of U

swap

Read out Memory

bull Assume dot with an electron with some information stored in spin-state

bull Connect two leads to dot

bull Apply a small bias (ΔV) rarr Current (i)Energy

position

Gate bias

Spin up - electron

Unoccupied state

i

Another Spin up electron enters dot

bull Pauli principle forces electrons with spin up to occupy the higher energy state

bull Negligible chance of tunneling

E

position

Gate bias

Spin up - electron

Higher energy level(forbidden classically)

i=0

Spin down electron enters dot

bull Pauli principle allows the new electron to join the same energy level as the original electron

bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow

E

position

Gate bias

Spin up - electron

Unoccupied state

ine0

Read out Memory

bull We have a way of measuring the spin state of an electron in a quantum dot

bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties

bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique

bull The electron current can be specialized (we can aim its spin to make measurement efficient)

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

The Physical System Excitons Trapped in GaAs Quantum Dots

bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid

bull Often formed when photons excite electrons from the valence band into the conduction band

bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses

bull Decay by radiating photons Decay time ~50ps-1ns

bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt

Initialization

bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to

prevent thermal excitations

bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the

register

Relatively Long Decoherence Times

bull Mechanismsndash Radiative Decay ~10ps-1ns

bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations

bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral

bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps

or less

A Universal set of Quantum Gates

bull Single Qubit Rotations through laser induced Rabi Oscillations

bull CNOT operations through dipole interactions and laser excitation

Single Qubit Gates Rabi Flopping

bull Light-particle interaction is characterized by the product of the dipole moment and the electric field

μbullE(t)= ħR(t)

Where R(t) is the Rabi frequency and the pulse area is given by

Θ(t)=intR(t)dt

and the state at time t is then given by

Cos(Θ2)|0gt+Sin(Θ2)|1gt

Stufler et al

Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses

Cos(Θ2)|0gt+Sin(Θ2)|1gt

|1gt =gt electric charge

=gtPhotocurrent (PC)PC~Sin2(Θ2)

π-pulse corresponds to a population inversion

CNOT Dipole Coupling

Nearest neighbor interactions alter the energy states

Effective energy Ersquoi = Ei + sumjnei ∆Eij nj

Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied

Overcoming Short Interaction Distances

bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs

bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots

bull Optical Cavity coupling and fiber optical interconnects have also been proposed

Read Out of Specified Qubit States

bull Optical readoutExcitons decay spontaneously and the resulting radiation can be

measured

Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD

Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination

In either case repeated measurements have to be made A single shot readout still needs to be developed

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

• Quantum Dots
• Implementing Quantum Computers
• What is a quantum dot
• Gallium Arsenide Quantum Dots
• Energy Band Levels
• Slide 6
• Slide 7
• Tuning Quantum Dots
• Manufacturing methods
• Electron Beam Lithography
• Slide 11
• Molecular Beam Epitaxy
• Spin Quantum Computing
• Decoherence time ~ 100ns
• Artificial Atom
• Adding Electrons changing Vgate
• Comparison with Hydrogen
• Tuning the Quantum Dot
• Single Qubit manipulation
• Two Qubit Manipulation
• Ground State Splitting (J = Et ndash Es)
• Solving J(B(t)) Exchange Coupling
• SWAP - gate
• XOR ~ CNOT
• Read out Memory
• Another Spin up electron enters dot
• Spin down electron enters dot
• Slide 28
• 5 DiVincenzo QC Criteria
• The Physical System Excitons Trapped in GaAs Quantum Dots
• Initialization
• Relatively Long Decoherence Times
• A Universal set of Quantum Gates
• Single Qubit Gates Rabi Flopping
• Stufler et al
• CNOT Dipole Coupling
• Overcoming Short Interaction Distances
• Read Out of Specified Qubit States
• Slide 39

Comparison with Hydrogenbull Artificial Atom

Energy levels ~ 1meV

Size ~ 10μm

Weak magnetic fields can affect energy levels

bull Hydrogen

Energy levels ~ 1eV

Size ~ 1Aring

Only strong magnetic fields can perturb energy levels

Factor 1000

Tuning the Quantum Dot

bull Tune so we have one valence electron

bull Initial state can be set by applying homogeneous magnetic field rarr |0gt

bull Low temperature kT lt ΔE (state gap)

bull Now we have defined our single qubit

Energy

position

Gate bias

Spin up - electron

Unoccupied state

Single Qubit manipulation

bull Unitary operations can be made by applying a local magnetic field H

ZE = -μB = g

μB SB

bull MF microscopebull AF microscopebull Sub grid of currentbull Magnetic dotsbull Etc

(Magnetic force microscope tip)

Two Qubit Manipulation

bull Complete set of logic requires a CNOT

bull Dots are placed so close that they overlap and interact

bull Hspin

= J(t)S1S

2

Exchange couplingJ(tEB) = E

triplet -E

singlet

(4th order Harmonic Oscillator)

Ground State Splitting (J = Et ndash E

s)

bull 2 coupled fermions must have an total anti-symmetric wave function

bull Lowest coupled state is the singlet It has a symmetric spatial wave function and an anti symmetric spin (Coulomb dominates)|ψ

sgt ~ (|12gt + |21gt) (|darruarrgt - |uarrdarrgt)

bull The triplet states are (|darrdarrgt)|ψ

tgt ~ (|12gt - |21gt) (|darruarrgt + |uarrdarrgt)

(|uarruarrgt)bull lt1|2gt ne 0 |igt is spatial wf Coulomb dominates

Solving J(B(t)) Exchange Coupling

bull Different solutions Heitler-London Hund-Mulliken Hubbard

bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations

SWAP - gatebull Assume J can be pulsed

J(t) = 0 J0

Formula 1

Formula 2

bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]

XOR ~ CNOT

bull Formula 3

bull Requirements Spin rotations about the z-axis Squareroot of U

swap

Read out Memory

bull Assume dot with an electron with some information stored in spin-state

bull Connect two leads to dot

bull Apply a small bias (ΔV) rarr Current (i)Energy

position

Gate bias

Spin up - electron

Unoccupied state

i

Another Spin up electron enters dot

bull Pauli principle forces electrons with spin up to occupy the higher energy state

bull Negligible chance of tunneling

E

position

Gate bias

Spin up - electron

Higher energy level(forbidden classically)

i=0

Spin down electron enters dot

bull Pauli principle allows the new electron to join the same energy level as the original electron

bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow

E

position

Gate bias

Spin up - electron

Unoccupied state

ine0

Read out Memory

bull We have a way of measuring the spin state of an electron in a quantum dot

bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties

bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique

bull The electron current can be specialized (we can aim its spin to make measurement efficient)

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

The Physical System Excitons Trapped in GaAs Quantum Dots

bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid

bull Often formed when photons excite electrons from the valence band into the conduction band

bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses

bull Decay by radiating photons Decay time ~50ps-1ns

bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt

Initialization

bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to

prevent thermal excitations

bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the

register

Relatively Long Decoherence Times

bull Mechanismsndash Radiative Decay ~10ps-1ns

bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations

bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral

bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps

or less

A Universal set of Quantum Gates

bull Single Qubit Rotations through laser induced Rabi Oscillations

bull CNOT operations through dipole interactions and laser excitation

Single Qubit Gates Rabi Flopping

bull Light-particle interaction is characterized by the product of the dipole moment and the electric field

μbullE(t)= ħR(t)

Where R(t) is the Rabi frequency and the pulse area is given by

Θ(t)=intR(t)dt

and the state at time t is then given by

Cos(Θ2)|0gt+Sin(Θ2)|1gt

Stufler et al

Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses

Cos(Θ2)|0gt+Sin(Θ2)|1gt

|1gt =gt electric charge

=gtPhotocurrent (PC)PC~Sin2(Θ2)

π-pulse corresponds to a population inversion

CNOT Dipole Coupling

Nearest neighbor interactions alter the energy states

Effective energy Ersquoi = Ei + sumjnei ∆Eij nj

Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied

Overcoming Short Interaction Distances

bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs

bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots

bull Optical Cavity coupling and fiber optical interconnects have also been proposed

Read Out of Specified Qubit States

bull Optical readoutExcitons decay spontaneously and the resulting radiation can be

measured

Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD

Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination

In either case repeated measurements have to be made A single shot readout still needs to be developed

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

• Quantum Dots
• Implementing Quantum Computers
• What is a quantum dot
• Gallium Arsenide Quantum Dots
• Energy Band Levels
• Slide 6
• Slide 7
• Tuning Quantum Dots
• Manufacturing methods
• Electron Beam Lithography
• Slide 11
• Molecular Beam Epitaxy
• Spin Quantum Computing
• Decoherence time ~ 100ns
• Artificial Atom
• Adding Electrons changing Vgate
• Comparison with Hydrogen
• Tuning the Quantum Dot
• Single Qubit manipulation
• Two Qubit Manipulation
• Ground State Splitting (J = Et ndash Es)
• Solving J(B(t)) Exchange Coupling
• SWAP - gate
• XOR ~ CNOT
• Read out Memory
• Another Spin up electron enters dot
• Spin down electron enters dot
• Slide 28
• 5 DiVincenzo QC Criteria
• The Physical System Excitons Trapped in GaAs Quantum Dots
• Initialization
• Relatively Long Decoherence Times
• A Universal set of Quantum Gates
• Single Qubit Gates Rabi Flopping
• Stufler et al
• CNOT Dipole Coupling
• Overcoming Short Interaction Distances
• Read Out of Specified Qubit States
• Slide 39

Tuning the Quantum Dot

bull Tune so we have one valence electron

bull Initial state can be set by applying homogeneous magnetic field rarr |0gt

bull Low temperature kT lt ΔE (state gap)

bull Now we have defined our single qubit

Energy

position

Gate bias

Spin up - electron

Unoccupied state

Single Qubit manipulation

bull Unitary operations can be made by applying a local magnetic field H

ZE = -μB = g

μB SB

bull MF microscopebull AF microscopebull Sub grid of currentbull Magnetic dotsbull Etc

(Magnetic force microscope tip)

Two Qubit Manipulation

bull Complete set of logic requires a CNOT

bull Dots are placed so close that they overlap and interact

bull Hspin

= J(t)S1S

2

Exchange couplingJ(tEB) = E

triplet -E

singlet

(4th order Harmonic Oscillator)

Ground State Splitting (J = Et ndash E

s)

bull 2 coupled fermions must have an total anti-symmetric wave function

bull Lowest coupled state is the singlet It has a symmetric spatial wave function and an anti symmetric spin (Coulomb dominates)|ψ

sgt ~ (|12gt + |21gt) (|darruarrgt - |uarrdarrgt)

bull The triplet states are (|darrdarrgt)|ψ

tgt ~ (|12gt - |21gt) (|darruarrgt + |uarrdarrgt)

(|uarruarrgt)bull lt1|2gt ne 0 |igt is spatial wf Coulomb dominates

Solving J(B(t)) Exchange Coupling

bull Different solutions Heitler-London Hund-Mulliken Hubbard

bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations

SWAP - gatebull Assume J can be pulsed

J(t) = 0 J0

Formula 1

Formula 2

bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]

XOR ~ CNOT

bull Formula 3

bull Requirements Spin rotations about the z-axis Squareroot of U

swap

Read out Memory

bull Assume dot with an electron with some information stored in spin-state

bull Connect two leads to dot

bull Apply a small bias (ΔV) rarr Current (i)Energy

position

Gate bias

Spin up - electron

Unoccupied state

i

Another Spin up electron enters dot

bull Pauli principle forces electrons with spin up to occupy the higher energy state

bull Negligible chance of tunneling

E

position

Gate bias

Spin up - electron

Higher energy level(forbidden classically)

i=0

Spin down electron enters dot

bull Pauli principle allows the new electron to join the same energy level as the original electron

bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow

E

position

Gate bias

Spin up - electron

Unoccupied state

ine0

Read out Memory

bull We have a way of measuring the spin state of an electron in a quantum dot

bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties

bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique

bull The electron current can be specialized (we can aim its spin to make measurement efficient)

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

The Physical System Excitons Trapped in GaAs Quantum Dots

bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid

bull Often formed when photons excite electrons from the valence band into the conduction band

bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses

bull Decay by radiating photons Decay time ~50ps-1ns

bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt

Initialization

bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to

prevent thermal excitations

bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the

register

Relatively Long Decoherence Times

bull Mechanismsndash Radiative Decay ~10ps-1ns

bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations

bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral

bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps

or less

A Universal set of Quantum Gates

bull Single Qubit Rotations through laser induced Rabi Oscillations

bull CNOT operations through dipole interactions and laser excitation

Single Qubit Gates Rabi Flopping

bull Light-particle interaction is characterized by the product of the dipole moment and the electric field

μbullE(t)= ħR(t)

Where R(t) is the Rabi frequency and the pulse area is given by

Θ(t)=intR(t)dt

and the state at time t is then given by

Cos(Θ2)|0gt+Sin(Θ2)|1gt

Stufler et al

Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses

Cos(Θ2)|0gt+Sin(Θ2)|1gt

|1gt =gt electric charge

=gtPhotocurrent (PC)PC~Sin2(Θ2)

π-pulse corresponds to a population inversion

CNOT Dipole Coupling

Nearest neighbor interactions alter the energy states

Effective energy Ersquoi = Ei + sumjnei ∆Eij nj

Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied

Overcoming Short Interaction Distances

bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs

bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots

bull Optical Cavity coupling and fiber optical interconnects have also been proposed

Read Out of Specified Qubit States

bull Optical readoutExcitons decay spontaneously and the resulting radiation can be

measured

Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD

Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination

In either case repeated measurements have to be made A single shot readout still needs to be developed

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

• Quantum Dots
• Implementing Quantum Computers
• What is a quantum dot
• Gallium Arsenide Quantum Dots
• Energy Band Levels
• Slide 6
• Slide 7
• Tuning Quantum Dots
• Manufacturing methods
• Electron Beam Lithography
• Slide 11
• Molecular Beam Epitaxy
• Spin Quantum Computing
• Decoherence time ~ 100ns
• Artificial Atom
• Adding Electrons changing Vgate
• Comparison with Hydrogen
• Tuning the Quantum Dot
• Single Qubit manipulation
• Two Qubit Manipulation
• Ground State Splitting (J = Et ndash Es)
• Solving J(B(t)) Exchange Coupling
• SWAP - gate
• XOR ~ CNOT
• Read out Memory
• Another Spin up electron enters dot
• Spin down electron enters dot
• Slide 28
• 5 DiVincenzo QC Criteria
• The Physical System Excitons Trapped in GaAs Quantum Dots
• Initialization
• Relatively Long Decoherence Times
• A Universal set of Quantum Gates
• Single Qubit Gates Rabi Flopping
• Stufler et al
• CNOT Dipole Coupling
• Overcoming Short Interaction Distances
• Read Out of Specified Qubit States
• Slide 39

Single Qubit manipulation

bull Unitary operations can be made by applying a local magnetic field H

ZE = -μB = g

μB SB

bull MF microscopebull AF microscopebull Sub grid of currentbull Magnetic dotsbull Etc

(Magnetic force microscope tip)

Two Qubit Manipulation

bull Complete set of logic requires a CNOT

bull Dots are placed so close that they overlap and interact

bull Hspin

= J(t)S1S

2

Exchange couplingJ(tEB) = E

triplet -E

singlet

(4th order Harmonic Oscillator)

Ground State Splitting (J = Et ndash E

s)

bull 2 coupled fermions must have an total anti-symmetric wave function

bull Lowest coupled state is the singlet It has a symmetric spatial wave function and an anti symmetric spin (Coulomb dominates)|ψ

sgt ~ (|12gt + |21gt) (|darruarrgt - |uarrdarrgt)

bull The triplet states are (|darrdarrgt)|ψ

tgt ~ (|12gt - |21gt) (|darruarrgt + |uarrdarrgt)

(|uarruarrgt)bull lt1|2gt ne 0 |igt is spatial wf Coulomb dominates

Solving J(B(t)) Exchange Coupling

bull Different solutions Heitler-London Hund-Mulliken Hubbard

bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations

SWAP - gatebull Assume J can be pulsed

J(t) = 0 J0

Formula 1

Formula 2

bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]

XOR ~ CNOT

bull Formula 3

bull Requirements Spin rotations about the z-axis Squareroot of U

swap

Read out Memory

bull Assume dot with an electron with some information stored in spin-state

bull Connect two leads to dot

bull Apply a small bias (ΔV) rarr Current (i)Energy

position

Gate bias

Spin up - electron

Unoccupied state

i

Another Spin up electron enters dot

bull Pauli principle forces electrons with spin up to occupy the higher energy state

bull Negligible chance of tunneling

E

position

Gate bias

Spin up - electron

Higher energy level(forbidden classically)

i=0

Spin down electron enters dot

bull Pauli principle allows the new electron to join the same energy level as the original electron

bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow

E

position

Gate bias

Spin up - electron

Unoccupied state

ine0

Read out Memory

bull We have a way of measuring the spin state of an electron in a quantum dot

bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties

bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique

bull The electron current can be specialized (we can aim its spin to make measurement efficient)

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

The Physical System Excitons Trapped in GaAs Quantum Dots

bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid

bull Often formed when photons excite electrons from the valence band into the conduction band

bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses

bull Decay by radiating photons Decay time ~50ps-1ns

bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt

Initialization

bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to

prevent thermal excitations

bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the

register

Relatively Long Decoherence Times

bull Mechanismsndash Radiative Decay ~10ps-1ns

bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations

bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral

bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps

or less

A Universal set of Quantum Gates

bull Single Qubit Rotations through laser induced Rabi Oscillations

bull CNOT operations through dipole interactions and laser excitation

Single Qubit Gates Rabi Flopping

bull Light-particle interaction is characterized by the product of the dipole moment and the electric field

μbullE(t)= ħR(t)

Where R(t) is the Rabi frequency and the pulse area is given by

Θ(t)=intR(t)dt

and the state at time t is then given by

Cos(Θ2)|0gt+Sin(Θ2)|1gt

Stufler et al

Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses

Cos(Θ2)|0gt+Sin(Θ2)|1gt

|1gt =gt electric charge

=gtPhotocurrent (PC)PC~Sin2(Θ2)

π-pulse corresponds to a population inversion

CNOT Dipole Coupling

Nearest neighbor interactions alter the energy states

Effective energy Ersquoi = Ei + sumjnei ∆Eij nj

Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied

Overcoming Short Interaction Distances

bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs

bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots

bull Optical Cavity coupling and fiber optical interconnects have also been proposed

Read Out of Specified Qubit States

bull Optical readoutExcitons decay spontaneously and the resulting radiation can be

measured

Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD

Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination

In either case repeated measurements have to be made A single shot readout still needs to be developed

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

• Quantum Dots
• Implementing Quantum Computers
• What is a quantum dot
• Gallium Arsenide Quantum Dots
• Energy Band Levels
• Slide 6
• Slide 7
• Tuning Quantum Dots
• Manufacturing methods
• Electron Beam Lithography
• Slide 11
• Molecular Beam Epitaxy
• Spin Quantum Computing
• Decoherence time ~ 100ns
• Artificial Atom
• Adding Electrons changing Vgate
• Comparison with Hydrogen
• Tuning the Quantum Dot
• Single Qubit manipulation
• Two Qubit Manipulation
• Ground State Splitting (J = Et ndash Es)
• Solving J(B(t)) Exchange Coupling
• SWAP - gate
• XOR ~ CNOT
• Read out Memory
• Another Spin up electron enters dot
• Spin down electron enters dot
• Slide 28
• 5 DiVincenzo QC Criteria
• The Physical System Excitons Trapped in GaAs Quantum Dots
• Initialization
• Relatively Long Decoherence Times
• A Universal set of Quantum Gates
• Single Qubit Gates Rabi Flopping
• Stufler et al
• CNOT Dipole Coupling
• Overcoming Short Interaction Distances
• Read Out of Specified Qubit States
• Slide 39

Two Qubit Manipulation

bull Complete set of logic requires a CNOT

bull Dots are placed so close that they overlap and interact

bull Hspin

= J(t)S1S

2

Exchange couplingJ(tEB) = E

triplet -E

singlet

(4th order Harmonic Oscillator)

Ground State Splitting (J = Et ndash E

s)

bull 2 coupled fermions must have an total anti-symmetric wave function

bull Lowest coupled state is the singlet It has a symmetric spatial wave function and an anti symmetric spin (Coulomb dominates)|ψ

sgt ~ (|12gt + |21gt) (|darruarrgt - |uarrdarrgt)

bull The triplet states are (|darrdarrgt)|ψ

tgt ~ (|12gt - |21gt) (|darruarrgt + |uarrdarrgt)

(|uarruarrgt)bull lt1|2gt ne 0 |igt is spatial wf Coulomb dominates

Solving J(B(t)) Exchange Coupling

bull Different solutions Heitler-London Hund-Mulliken Hubbard

bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations

SWAP - gatebull Assume J can be pulsed

J(t) = 0 J0

Formula 1

Formula 2

bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]

XOR ~ CNOT

bull Formula 3

bull Requirements Spin rotations about the z-axis Squareroot of U

swap

Read out Memory

bull Assume dot with an electron with some information stored in spin-state

bull Connect two leads to dot

bull Apply a small bias (ΔV) rarr Current (i)Energy

position

Gate bias

Spin up - electron

Unoccupied state

i

Another Spin up electron enters dot

bull Pauli principle forces electrons with spin up to occupy the higher energy state

bull Negligible chance of tunneling

E

position

Gate bias

Spin up - electron

Higher energy level(forbidden classically)

i=0

Spin down electron enters dot

bull Pauli principle allows the new electron to join the same energy level as the original electron

bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow

E

position

Gate bias

Spin up - electron

Unoccupied state

ine0

Read out Memory

bull We have a way of measuring the spin state of an electron in a quantum dot

bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties

bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique

bull The electron current can be specialized (we can aim its spin to make measurement efficient)

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

The Physical System Excitons Trapped in GaAs Quantum Dots

bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid

bull Often formed when photons excite electrons from the valence band into the conduction band

bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses

bull Decay by radiating photons Decay time ~50ps-1ns

bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt

Initialization

bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to

prevent thermal excitations

bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the

register

Relatively Long Decoherence Times

bull Mechanismsndash Radiative Decay ~10ps-1ns

bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations

bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral

bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps

or less

A Universal set of Quantum Gates

bull Single Qubit Rotations through laser induced Rabi Oscillations

bull CNOT operations through dipole interactions and laser excitation

Single Qubit Gates Rabi Flopping

bull Light-particle interaction is characterized by the product of the dipole moment and the electric field

μbullE(t)= ħR(t)

Where R(t) is the Rabi frequency and the pulse area is given by

Θ(t)=intR(t)dt

and the state at time t is then given by

Cos(Θ2)|0gt+Sin(Θ2)|1gt

Stufler et al

Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses

Cos(Θ2)|0gt+Sin(Θ2)|1gt

|1gt =gt electric charge

=gtPhotocurrent (PC)PC~Sin2(Θ2)

π-pulse corresponds to a population inversion

CNOT Dipole Coupling

Nearest neighbor interactions alter the energy states

Effective energy Ersquoi = Ei + sumjnei ∆Eij nj

Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied

Overcoming Short Interaction Distances

bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs

bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots

bull Optical Cavity coupling and fiber optical interconnects have also been proposed

Read Out of Specified Qubit States

bull Optical readoutExcitons decay spontaneously and the resulting radiation can be

measured

Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD

Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination

In either case repeated measurements have to be made A single shot readout still needs to be developed

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

• Quantum Dots
• Implementing Quantum Computers
• What is a quantum dot
• Gallium Arsenide Quantum Dots
• Energy Band Levels
• Slide 6
• Slide 7
• Tuning Quantum Dots
• Manufacturing methods
• Electron Beam Lithography
• Slide 11
• Molecular Beam Epitaxy
• Spin Quantum Computing
• Decoherence time ~ 100ns
• Artificial Atom
• Adding Electrons changing Vgate
• Comparison with Hydrogen
• Tuning the Quantum Dot
• Single Qubit manipulation
• Two Qubit Manipulation
• Ground State Splitting (J = Et ndash Es)
• Solving J(B(t)) Exchange Coupling
• SWAP - gate
• XOR ~ CNOT
• Read out Memory
• Another Spin up electron enters dot
• Spin down electron enters dot
• Slide 28
• 5 DiVincenzo QC Criteria
• The Physical System Excitons Trapped in GaAs Quantum Dots
• Initialization
• Relatively Long Decoherence Times
• A Universal set of Quantum Gates
• Single Qubit Gates Rabi Flopping
• Stufler et al
• CNOT Dipole Coupling
• Overcoming Short Interaction Distances
• Read Out of Specified Qubit States
• Slide 39

Ground State Splitting (J = Et ndash E

s)

bull 2 coupled fermions must have an total anti-symmetric wave function

bull Lowest coupled state is the singlet It has a symmetric spatial wave function and an anti symmetric spin (Coulomb dominates)|ψ

sgt ~ (|12gt + |21gt) (|darruarrgt - |uarrdarrgt)

bull The triplet states are (|darrdarrgt)|ψ

tgt ~ (|12gt - |21gt) (|darruarrgt + |uarrdarrgt)

(|uarruarrgt)bull lt1|2gt ne 0 |igt is spatial wf Coulomb dominates

Solving J(B(t)) Exchange Coupling

bull Different solutions Heitler-London Hund-Mulliken Hubbard

bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations

SWAP - gatebull Assume J can be pulsed

J(t) = 0 J0

Formula 1

Formula 2

bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]

XOR ~ CNOT

bull Formula 3

bull Requirements Spin rotations about the z-axis Squareroot of U

swap

Read out Memory

bull Assume dot with an electron with some information stored in spin-state

bull Connect two leads to dot

bull Apply a small bias (ΔV) rarr Current (i)Energy

position

Gate bias

Spin up - electron

Unoccupied state

i

Another Spin up electron enters dot

bull Pauli principle forces electrons with spin up to occupy the higher energy state

bull Negligible chance of tunneling

E

position

Gate bias

Spin up - electron

Higher energy level(forbidden classically)

i=0

Spin down electron enters dot

bull Pauli principle allows the new electron to join the same energy level as the original electron

bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow

E

position

Gate bias

Spin up - electron

Unoccupied state

ine0

Read out Memory

bull We have a way of measuring the spin state of an electron in a quantum dot

bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties

bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique

bull The electron current can be specialized (we can aim its spin to make measurement efficient)

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

The Physical System Excitons Trapped in GaAs Quantum Dots

bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid

bull Often formed when photons excite electrons from the valence band into the conduction band

bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses

bull Decay by radiating photons Decay time ~50ps-1ns

bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt

Initialization

bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to

prevent thermal excitations

bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the

register

Relatively Long Decoherence Times

bull Mechanismsndash Radiative Decay ~10ps-1ns

bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations

bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral

bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps

or less

A Universal set of Quantum Gates

bull Single Qubit Rotations through laser induced Rabi Oscillations

bull CNOT operations through dipole interactions and laser excitation

Single Qubit Gates Rabi Flopping

bull Light-particle interaction is characterized by the product of the dipole moment and the electric field

μbullE(t)= ħR(t)

Where R(t) is the Rabi frequency and the pulse area is given by

Θ(t)=intR(t)dt

and the state at time t is then given by

Cos(Θ2)|0gt+Sin(Θ2)|1gt

Stufler et al

Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses

Cos(Θ2)|0gt+Sin(Θ2)|1gt

|1gt =gt electric charge

=gtPhotocurrent (PC)PC~Sin2(Θ2)

π-pulse corresponds to a population inversion

CNOT Dipole Coupling

Nearest neighbor interactions alter the energy states

Effective energy Ersquoi = Ei + sumjnei ∆Eij nj

Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied

Overcoming Short Interaction Distances

bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs

bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots

bull Optical Cavity coupling and fiber optical interconnects have also been proposed

Read Out of Specified Qubit States

bull Optical readoutExcitons decay spontaneously and the resulting radiation can be

measured

Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD

Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination

In either case repeated measurements have to be made A single shot readout still needs to be developed

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

• Quantum Dots
• Implementing Quantum Computers
• What is a quantum dot
• Gallium Arsenide Quantum Dots
• Energy Band Levels
• Slide 6
• Slide 7
• Tuning Quantum Dots
• Manufacturing methods
• Electron Beam Lithography
• Slide 11
• Molecular Beam Epitaxy
• Spin Quantum Computing
• Decoherence time ~ 100ns
• Artificial Atom
• Adding Electrons changing Vgate
• Comparison with Hydrogen
• Tuning the Quantum Dot
• Single Qubit manipulation
• Two Qubit Manipulation
• Ground State Splitting (J = Et ndash Es)
• Solving J(B(t)) Exchange Coupling
• SWAP - gate
• XOR ~ CNOT
• Read out Memory
• Another Spin up electron enters dot
• Spin down electron enters dot
• Slide 28
• 5 DiVincenzo QC Criteria
• The Physical System Excitons Trapped in GaAs Quantum Dots
• Initialization
• Relatively Long Decoherence Times
• A Universal set of Quantum Gates
• Single Qubit Gates Rabi Flopping
• Stufler et al
• CNOT Dipole Coupling
• Overcoming Short Interaction Distances
• Read Out of Specified Qubit States
• Slide 39

Solving J(B(t)) Exchange Coupling

bull Different solutions Heitler-London Hund-Mulliken Hubbard

bull Important conclusionWe can control coupling from zero to non-zero by changing the magnetic field rarr We can perform two qubit operations

SWAP - gatebull Assume J can be pulsed

J(t) = 0 J0

Formula 1

Formula 2

bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]

XOR ~ CNOT

bull Formula 3

bull Requirements Spin rotations about the z-axis Squareroot of U

swap

Read out Memory

bull Assume dot with an electron with some information stored in spin-state

bull Connect two leads to dot

bull Apply a small bias (ΔV) rarr Current (i)Energy

position

Gate bias

Spin up - electron

Unoccupied state

i

Another Spin up electron enters dot

bull Pauli principle forces electrons with spin up to occupy the higher energy state

bull Negligible chance of tunneling

E

position

Gate bias

Spin up - electron

Higher energy level(forbidden classically)

i=0

Spin down electron enters dot

bull Pauli principle allows the new electron to join the same energy level as the original electron

bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow

E

position

Gate bias

Spin up - electron

Unoccupied state

ine0

Read out Memory

bull We have a way of measuring the spin state of an electron in a quantum dot

bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties

bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique

bull The electron current can be specialized (we can aim its spin to make measurement efficient)

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

The Physical System Excitons Trapped in GaAs Quantum Dots

bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid

bull Often formed when photons excite electrons from the valence band into the conduction band

bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses

bull Decay by radiating photons Decay time ~50ps-1ns

bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt

Initialization

bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to

prevent thermal excitations

bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the

register

Relatively Long Decoherence Times

bull Mechanismsndash Radiative Decay ~10ps-1ns

bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations

bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral

bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps

or less

A Universal set of Quantum Gates

bull Single Qubit Rotations through laser induced Rabi Oscillations

bull CNOT operations through dipole interactions and laser excitation

Single Qubit Gates Rabi Flopping

bull Light-particle interaction is characterized by the product of the dipole moment and the electric field

μbullE(t)= ħR(t)

Where R(t) is the Rabi frequency and the pulse area is given by

Θ(t)=intR(t)dt

and the state at time t is then given by

Cos(Θ2)|0gt+Sin(Θ2)|1gt

Stufler et al

Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses

Cos(Θ2)|0gt+Sin(Θ2)|1gt

|1gt =gt electric charge

=gtPhotocurrent (PC)PC~Sin2(Θ2)

π-pulse corresponds to a population inversion

CNOT Dipole Coupling

Nearest neighbor interactions alter the energy states

Effective energy Ersquoi = Ei + sumjnei ∆Eij nj

Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied

Overcoming Short Interaction Distances

bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs

bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots

bull Optical Cavity coupling and fiber optical interconnects have also been proposed

Read Out of Specified Qubit States

bull Optical readoutExcitons decay spontaneously and the resulting radiation can be

measured

Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD

Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination

In either case repeated measurements have to be made A single shot readout still needs to be developed

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

• Quantum Dots
• Implementing Quantum Computers
• What is a quantum dot
• Gallium Arsenide Quantum Dots
• Energy Band Levels
• Slide 6
• Slide 7
• Tuning Quantum Dots
• Manufacturing methods
• Electron Beam Lithography
• Slide 11
• Molecular Beam Epitaxy
• Spin Quantum Computing
• Decoherence time ~ 100ns
• Artificial Atom
• Adding Electrons changing Vgate
• Comparison with Hydrogen
• Tuning the Quantum Dot
• Single Qubit manipulation
• Two Qubit Manipulation
• Ground State Splitting (J = Et ndash Es)
• Solving J(B(t)) Exchange Coupling
• SWAP - gate
• XOR ~ CNOT
• Read out Memory
• Another Spin up electron enters dot
• Spin down electron enters dot
• Slide 28
• 5 DiVincenzo QC Criteria
• The Physical System Excitons Trapped in GaAs Quantum Dots
• Initialization
• Relatively Long Decoherence Times
• A Universal set of Quantum Gates
• Single Qubit Gates Rabi Flopping
• Stufler et al
• CNOT Dipole Coupling
• Overcoming Short Interaction Distances
• Read Out of Specified Qubit States
• Slide 39

SWAP - gatebull Assume J can be pulsed

J(t) = 0 J0

Formula 1

Formula 2

bull Now we can put many qubits on a line and move them so that they all can interact [not all at once though]

XOR ~ CNOT

bull Formula 3

bull Requirements Spin rotations about the z-axis Squareroot of U

swap

Read out Memory

bull Assume dot with an electron with some information stored in spin-state

bull Connect two leads to dot

bull Apply a small bias (ΔV) rarr Current (i)Energy

position

Gate bias

Spin up - electron

Unoccupied state

i

Another Spin up electron enters dot

bull Pauli principle forces electrons with spin up to occupy the higher energy state

bull Negligible chance of tunneling

E

position

Gate bias

Spin up - electron

Higher energy level(forbidden classically)

i=0

Spin down electron enters dot

bull Pauli principle allows the new electron to join the same energy level as the original electron

bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow

E

position

Gate bias

Spin up - electron

Unoccupied state

ine0

Read out Memory

bull We have a way of measuring the spin state of an electron in a quantum dot

bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties

bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique

bull The electron current can be specialized (we can aim its spin to make measurement efficient)

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

The Physical System Excitons Trapped in GaAs Quantum Dots

bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid

bull Often formed when photons excite electrons from the valence band into the conduction band

bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses

bull Decay by radiating photons Decay time ~50ps-1ns

bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt

Initialization

bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to

prevent thermal excitations

bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the

register

Relatively Long Decoherence Times

bull Mechanismsndash Radiative Decay ~10ps-1ns

bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations

bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral

bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps

or less

A Universal set of Quantum Gates

bull Single Qubit Rotations through laser induced Rabi Oscillations

bull CNOT operations through dipole interactions and laser excitation

Single Qubit Gates Rabi Flopping

bull Light-particle interaction is characterized by the product of the dipole moment and the electric field

μbullE(t)= ħR(t)

Where R(t) is the Rabi frequency and the pulse area is given by

Θ(t)=intR(t)dt

and the state at time t is then given by

Cos(Θ2)|0gt+Sin(Θ2)|1gt

Stufler et al

Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses

Cos(Θ2)|0gt+Sin(Θ2)|1gt

|1gt =gt electric charge

=gtPhotocurrent (PC)PC~Sin2(Θ2)

π-pulse corresponds to a population inversion

CNOT Dipole Coupling

Nearest neighbor interactions alter the energy states

Effective energy Ersquoi = Ei + sumjnei ∆Eij nj

Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied

Overcoming Short Interaction Distances

bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs

bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots

bull Optical Cavity coupling and fiber optical interconnects have also been proposed

Read Out of Specified Qubit States

bull Optical readoutExcitons decay spontaneously and the resulting radiation can be

measured

Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD

Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination

In either case repeated measurements have to be made A single shot readout still needs to be developed

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

• Quantum Dots
• Implementing Quantum Computers
• What is a quantum dot
• Gallium Arsenide Quantum Dots
• Energy Band Levels
• Slide 6
• Slide 7
• Tuning Quantum Dots
• Manufacturing methods
• Electron Beam Lithography
• Slide 11
• Molecular Beam Epitaxy
• Spin Quantum Computing
• Decoherence time ~ 100ns
• Artificial Atom
• Adding Electrons changing Vgate
• Comparison with Hydrogen
• Tuning the Quantum Dot
• Single Qubit manipulation
• Two Qubit Manipulation
• Ground State Splitting (J = Et ndash Es)
• Solving J(B(t)) Exchange Coupling
• SWAP - gate
• XOR ~ CNOT
• Read out Memory
• Another Spin up electron enters dot
• Spin down electron enters dot
• Slide 28
• 5 DiVincenzo QC Criteria
• The Physical System Excitons Trapped in GaAs Quantum Dots
• Initialization
• Relatively Long Decoherence Times
• A Universal set of Quantum Gates
• Single Qubit Gates Rabi Flopping
• Stufler et al
• CNOT Dipole Coupling
• Overcoming Short Interaction Distances
• Read Out of Specified Qubit States
• Slide 39

XOR ~ CNOT

bull Formula 3

bull Requirements Spin rotations about the z-axis Squareroot of U

swap

Read out Memory

bull Assume dot with an electron with some information stored in spin-state

bull Connect two leads to dot

bull Apply a small bias (ΔV) rarr Current (i)Energy

position

Gate bias

Spin up - electron

Unoccupied state

i

Another Spin up electron enters dot

bull Pauli principle forces electrons with spin up to occupy the higher energy state

bull Negligible chance of tunneling

E

position

Gate bias

Spin up - electron

Higher energy level(forbidden classically)

i=0

Spin down electron enters dot

bull Pauli principle allows the new electron to join the same energy level as the original electron

bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow

E

position

Gate bias

Spin up - electron

Unoccupied state

ine0

Read out Memory

bull We have a way of measuring the spin state of an electron in a quantum dot

bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties

bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique

bull The electron current can be specialized (we can aim its spin to make measurement efficient)

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

The Physical System Excitons Trapped in GaAs Quantum Dots

bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid

bull Often formed when photons excite electrons from the valence band into the conduction band

bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses

bull Decay by radiating photons Decay time ~50ps-1ns

bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt

Initialization

bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to

prevent thermal excitations

bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the

register

Relatively Long Decoherence Times

bull Mechanismsndash Radiative Decay ~10ps-1ns

bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations

bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral

bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps

or less

A Universal set of Quantum Gates

bull Single Qubit Rotations through laser induced Rabi Oscillations

bull CNOT operations through dipole interactions and laser excitation

Single Qubit Gates Rabi Flopping

bull Light-particle interaction is characterized by the product of the dipole moment and the electric field

μbullE(t)= ħR(t)

Where R(t) is the Rabi frequency and the pulse area is given by

Θ(t)=intR(t)dt

and the state at time t is then given by

Cos(Θ2)|0gt+Sin(Θ2)|1gt

Stufler et al

Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses

Cos(Θ2)|0gt+Sin(Θ2)|1gt

|1gt =gt electric charge

=gtPhotocurrent (PC)PC~Sin2(Θ2)

π-pulse corresponds to a population inversion

CNOT Dipole Coupling

Nearest neighbor interactions alter the energy states

Effective energy Ersquoi = Ei + sumjnei ∆Eij nj

Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied

Overcoming Short Interaction Distances

bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs

bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots

bull Optical Cavity coupling and fiber optical interconnects have also been proposed

Read Out of Specified Qubit States

bull Optical readoutExcitons decay spontaneously and the resulting radiation can be

measured

Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD

Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination

In either case repeated measurements have to be made A single shot readout still needs to be developed

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

• Quantum Dots
• Implementing Quantum Computers
• What is a quantum dot
• Gallium Arsenide Quantum Dots
• Energy Band Levels
• Slide 6
• Slide 7
• Tuning Quantum Dots
• Manufacturing methods
• Electron Beam Lithography
• Slide 11
• Molecular Beam Epitaxy
• Spin Quantum Computing
• Decoherence time ~ 100ns
• Artificial Atom
• Adding Electrons changing Vgate
• Comparison with Hydrogen
• Tuning the Quantum Dot
• Single Qubit manipulation
• Two Qubit Manipulation
• Ground State Splitting (J = Et ndash Es)
• Solving J(B(t)) Exchange Coupling
• SWAP - gate
• XOR ~ CNOT
• Read out Memory
• Another Spin up electron enters dot
• Spin down electron enters dot
• Slide 28
• 5 DiVincenzo QC Criteria
• The Physical System Excitons Trapped in GaAs Quantum Dots
• Initialization
• Relatively Long Decoherence Times
• A Universal set of Quantum Gates
• Single Qubit Gates Rabi Flopping
• Stufler et al
• CNOT Dipole Coupling
• Overcoming Short Interaction Distances
• Read Out of Specified Qubit States
• Slide 39

Read out Memory

bull Assume dot with an electron with some information stored in spin-state

bull Connect two leads to dot

bull Apply a small bias (ΔV) rarr Current (i)Energy

position

Gate bias

Spin up - electron

Unoccupied state

i

Another Spin up electron enters dot

bull Pauli principle forces electrons with spin up to occupy the higher energy state

bull Negligible chance of tunneling

E

position

Gate bias

Spin up - electron

Higher energy level(forbidden classically)

i=0

Spin down electron enters dot

bull Pauli principle allows the new electron to join the same energy level as the original electron

bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow

E

position

Gate bias

Spin up - electron

Unoccupied state

ine0

Read out Memory

bull We have a way of measuring the spin state of an electron in a quantum dot

bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties

bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique

bull The electron current can be specialized (we can aim its spin to make measurement efficient)

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

The Physical System Excitons Trapped in GaAs Quantum Dots

bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid

bull Often formed when photons excite electrons from the valence band into the conduction band

bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses

bull Decay by radiating photons Decay time ~50ps-1ns

bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt

Initialization

bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to

prevent thermal excitations

bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the

register

Relatively Long Decoherence Times

bull Mechanismsndash Radiative Decay ~10ps-1ns

bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations

bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral

bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps

or less

A Universal set of Quantum Gates

bull Single Qubit Rotations through laser induced Rabi Oscillations

bull CNOT operations through dipole interactions and laser excitation

Single Qubit Gates Rabi Flopping

bull Light-particle interaction is characterized by the product of the dipole moment and the electric field

μbullE(t)= ħR(t)

Where R(t) is the Rabi frequency and the pulse area is given by

Θ(t)=intR(t)dt

and the state at time t is then given by

Cos(Θ2)|0gt+Sin(Θ2)|1gt

Stufler et al

Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses

Cos(Θ2)|0gt+Sin(Θ2)|1gt

|1gt =gt electric charge

=gtPhotocurrent (PC)PC~Sin2(Θ2)

π-pulse corresponds to a population inversion

CNOT Dipole Coupling

Nearest neighbor interactions alter the energy states

Effective energy Ersquoi = Ei + sumjnei ∆Eij nj

Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied

Overcoming Short Interaction Distances

bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs

bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots

bull Optical Cavity coupling and fiber optical interconnects have also been proposed

Read Out of Specified Qubit States

bull Optical readoutExcitons decay spontaneously and the resulting radiation can be

measured

Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD

Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination

In either case repeated measurements have to be made A single shot readout still needs to be developed

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

• Quantum Dots
• Implementing Quantum Computers
• What is a quantum dot
• Gallium Arsenide Quantum Dots
• Energy Band Levels
• Slide 6
• Slide 7
• Tuning Quantum Dots
• Manufacturing methods
• Electron Beam Lithography
• Slide 11
• Molecular Beam Epitaxy
• Spin Quantum Computing
• Decoherence time ~ 100ns
• Artificial Atom
• Adding Electrons changing Vgate
• Comparison with Hydrogen
• Tuning the Quantum Dot
• Single Qubit manipulation
• Two Qubit Manipulation
• Ground State Splitting (J = Et ndash Es)
• Solving J(B(t)) Exchange Coupling
• SWAP - gate
• XOR ~ CNOT
• Read out Memory
• Another Spin up electron enters dot
• Spin down electron enters dot
• Slide 28
• 5 DiVincenzo QC Criteria
• The Physical System Excitons Trapped in GaAs Quantum Dots
• Initialization
• Relatively Long Decoherence Times
• A Universal set of Quantum Gates
• Single Qubit Gates Rabi Flopping
• Stufler et al
• CNOT Dipole Coupling
• Overcoming Short Interaction Distances
• Read Out of Specified Qubit States
• Slide 39

Another Spin up electron enters dot

bull Pauli principle forces electrons with spin up to occupy the higher energy state

bull Negligible chance of tunneling

E

position

Gate bias

Spin up - electron

Higher energy level(forbidden classically)

i=0

Spin down electron enters dot

bull Pauli principle allows the new electron to join the same energy level as the original electron

bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow

E

position

Gate bias

Spin up - electron

Unoccupied state

ine0

Read out Memory

bull We have a way of measuring the spin state of an electron in a quantum dot

bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties

bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique

bull The electron current can be specialized (we can aim its spin to make measurement efficient)

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

The Physical System Excitons Trapped in GaAs Quantum Dots

bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid

bull Often formed when photons excite electrons from the valence band into the conduction band

bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses

bull Decay by radiating photons Decay time ~50ps-1ns

bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt

Initialization

bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to

prevent thermal excitations

bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the

register

Relatively Long Decoherence Times

bull Mechanismsndash Radiative Decay ~10ps-1ns

bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations

bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral

bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps

or less

A Universal set of Quantum Gates

bull Single Qubit Rotations through laser induced Rabi Oscillations

bull CNOT operations through dipole interactions and laser excitation

Single Qubit Gates Rabi Flopping

bull Light-particle interaction is characterized by the product of the dipole moment and the electric field

μbullE(t)= ħR(t)

Where R(t) is the Rabi frequency and the pulse area is given by

Θ(t)=intR(t)dt

and the state at time t is then given by

Cos(Θ2)|0gt+Sin(Θ2)|1gt

Stufler et al

Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses

Cos(Θ2)|0gt+Sin(Θ2)|1gt

|1gt =gt electric charge

=gtPhotocurrent (PC)PC~Sin2(Θ2)

π-pulse corresponds to a population inversion

CNOT Dipole Coupling

Nearest neighbor interactions alter the energy states

Effective energy Ersquoi = Ei + sumjnei ∆Eij nj

Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied

Overcoming Short Interaction Distances

bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs

bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots

bull Optical Cavity coupling and fiber optical interconnects have also been proposed

Read Out of Specified Qubit States

bull Optical readoutExcitons decay spontaneously and the resulting radiation can be

measured

Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD

Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination

In either case repeated measurements have to be made A single shot readout still needs to be developed

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

• Quantum Dots
• Implementing Quantum Computers
• What is a quantum dot
• Gallium Arsenide Quantum Dots
• Energy Band Levels
• Slide 6
• Slide 7
• Tuning Quantum Dots
• Manufacturing methods
• Electron Beam Lithography
• Slide 11
• Molecular Beam Epitaxy
• Spin Quantum Computing
• Decoherence time ~ 100ns
• Artificial Atom
• Adding Electrons changing Vgate
• Comparison with Hydrogen
• Tuning the Quantum Dot
• Single Qubit manipulation
• Two Qubit Manipulation
• Ground State Splitting (J = Et ndash Es)
• Solving J(B(t)) Exchange Coupling
• SWAP - gate
• XOR ~ CNOT
• Read out Memory
• Another Spin up electron enters dot
• Spin down electron enters dot
• Slide 28
• 5 DiVincenzo QC Criteria
• The Physical System Excitons Trapped in GaAs Quantum Dots
• Initialization
• Relatively Long Decoherence Times
• A Universal set of Quantum Gates
• Single Qubit Gates Rabi Flopping
• Stufler et al
• CNOT Dipole Coupling
• Overcoming Short Interaction Distances
• Read Out of Specified Qubit States
• Slide 39

Spin down electron enters dot

bull Pauli principle allows the new electron to join the same energy level as the original electron

bull Coulomb interaction perturbs the ground-state so that it is raised above the right bias and current will flow

E

position

Gate bias

Spin up - electron

Unoccupied state

ine0

Read out Memory

bull We have a way of measuring the spin state of an electron in a quantum dot

bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties

bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique

bull The electron current can be specialized (we can aim its spin to make measurement efficient)

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

The Physical System Excitons Trapped in GaAs Quantum Dots

bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid

bull Often formed when photons excite electrons from the valence band into the conduction band

bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses

bull Decay by radiating photons Decay time ~50ps-1ns

bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt

Initialization

bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to

prevent thermal excitations

bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the

register

Relatively Long Decoherence Times

bull Mechanismsndash Radiative Decay ~10ps-1ns

bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations

bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral

bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps

or less

A Universal set of Quantum Gates

bull Single Qubit Rotations through laser induced Rabi Oscillations

bull CNOT operations through dipole interactions and laser excitation

Single Qubit Gates Rabi Flopping

bull Light-particle interaction is characterized by the product of the dipole moment and the electric field

μbullE(t)= ħR(t)

Where R(t) is the Rabi frequency and the pulse area is given by

Θ(t)=intR(t)dt

and the state at time t is then given by

Cos(Θ2)|0gt+Sin(Θ2)|1gt

Stufler et al

Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses

Cos(Θ2)|0gt+Sin(Θ2)|1gt

|1gt =gt electric charge

=gtPhotocurrent (PC)PC~Sin2(Θ2)

π-pulse corresponds to a population inversion

CNOT Dipole Coupling

Nearest neighbor interactions alter the energy states

Effective energy Ersquoi = Ei + sumjnei ∆Eij nj

Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied

Overcoming Short Interaction Distances

bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs

bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots

bull Optical Cavity coupling and fiber optical interconnects have also been proposed

Read Out of Specified Qubit States

bull Optical readoutExcitons decay spontaneously and the resulting radiation can be

measured

Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD

Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination

In either case repeated measurements have to be made A single shot readout still needs to be developed

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

• Quantum Dots
• Implementing Quantum Computers
• What is a quantum dot
• Gallium Arsenide Quantum Dots
• Energy Band Levels
• Slide 6
• Slide 7
• Tuning Quantum Dots
• Manufacturing methods
• Electron Beam Lithography
• Slide 11
• Molecular Beam Epitaxy
• Spin Quantum Computing
• Decoherence time ~ 100ns
• Artificial Atom
• Adding Electrons changing Vgate
• Comparison with Hydrogen
• Tuning the Quantum Dot
• Single Qubit manipulation
• Two Qubit Manipulation
• Ground State Splitting (J = Et ndash Es)
• Solving J(B(t)) Exchange Coupling
• SWAP - gate
• XOR ~ CNOT
• Read out Memory
• Another Spin up electron enters dot
• Spin down electron enters dot
• Slide 28
• 5 DiVincenzo QC Criteria
• The Physical System Excitons Trapped in GaAs Quantum Dots
• Initialization
• Relatively Long Decoherence Times
• A Universal set of Quantum Gates
• Single Qubit Gates Rabi Flopping
• Stufler et al
• CNOT Dipole Coupling
• Overcoming Short Interaction Distances
• Read Out of Specified Qubit States
• Slide 39

Read out Memory

bull We have a way of measuring the spin state of an electron in a quantum dot

bull The first electron that passes though measures the spin-state in the dot and other electrons that follow will all have the same spin properties

bull To be able to predict the original state of the dot the state has to be prepared again and then measured using the same technique

bull The electron current can be specialized (we can aim its spin to make measurement efficient)

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

The Physical System Excitons Trapped in GaAs Quantum Dots

bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid

bull Often formed when photons excite electrons from the valence band into the conduction band

bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses

bull Decay by radiating photons Decay time ~50ps-1ns

bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt

Initialization

bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to

prevent thermal excitations

bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the

register

Relatively Long Decoherence Times

bull Mechanismsndash Radiative Decay ~10ps-1ns

bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations

bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral

bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps

or less

A Universal set of Quantum Gates

bull Single Qubit Rotations through laser induced Rabi Oscillations

bull CNOT operations through dipole interactions and laser excitation

Single Qubit Gates Rabi Flopping

bull Light-particle interaction is characterized by the product of the dipole moment and the electric field

μbullE(t)= ħR(t)

Where R(t) is the Rabi frequency and the pulse area is given by

Θ(t)=intR(t)dt

and the state at time t is then given by

Cos(Θ2)|0gt+Sin(Θ2)|1gt

Stufler et al

Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses

Cos(Θ2)|0gt+Sin(Θ2)|1gt

|1gt =gt electric charge

=gtPhotocurrent (PC)PC~Sin2(Θ2)

π-pulse corresponds to a population inversion

CNOT Dipole Coupling

Nearest neighbor interactions alter the energy states

Effective energy Ersquoi = Ei + sumjnei ∆Eij nj

Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied

Overcoming Short Interaction Distances

bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs

bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots

bull Optical Cavity coupling and fiber optical interconnects have also been proposed

Read Out of Specified Qubit States

bull Optical readoutExcitons decay spontaneously and the resulting radiation can be

measured

Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD

Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination

In either case repeated measurements have to be made A single shot readout still needs to be developed

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

• Quantum Dots
• Implementing Quantum Computers
• What is a quantum dot
• Gallium Arsenide Quantum Dots
• Energy Band Levels
• Slide 6
• Slide 7
• Tuning Quantum Dots
• Manufacturing methods
• Electron Beam Lithography
• Slide 11
• Molecular Beam Epitaxy
• Spin Quantum Computing
• Decoherence time ~ 100ns
• Artificial Atom
• Adding Electrons changing Vgate
• Comparison with Hydrogen
• Tuning the Quantum Dot
• Single Qubit manipulation
• Two Qubit Manipulation
• Ground State Splitting (J = Et ndash Es)
• Solving J(B(t)) Exchange Coupling
• SWAP - gate
• XOR ~ CNOT
• Read out Memory
• Another Spin up electron enters dot
• Spin down electron enters dot
• Slide 28
• 5 DiVincenzo QC Criteria
• The Physical System Excitons Trapped in GaAs Quantum Dots
• Initialization
• Relatively Long Decoherence Times
• A Universal set of Quantum Gates
• Single Qubit Gates Rabi Flopping
• Stufler et al
• CNOT Dipole Coupling
• Overcoming Short Interaction Distances
• Read Out of Specified Qubit States
• Slide 39

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

The Physical System Excitons Trapped in GaAs Quantum Dots

bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid

bull Often formed when photons excite electrons from the valence band into the conduction band

bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses

bull Decay by radiating photons Decay time ~50ps-1ns

bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt

Initialization

bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to

prevent thermal excitations

bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the

register

Relatively Long Decoherence Times

bull Mechanismsndash Radiative Decay ~10ps-1ns

bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations

bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral

bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps

or less

A Universal set of Quantum Gates

bull Single Qubit Rotations through laser induced Rabi Oscillations

bull CNOT operations through dipole interactions and laser excitation

Single Qubit Gates Rabi Flopping

bull Light-particle interaction is characterized by the product of the dipole moment and the electric field

μbullE(t)= ħR(t)

Where R(t) is the Rabi frequency and the pulse area is given by

Θ(t)=intR(t)dt

and the state at time t is then given by

Cos(Θ2)|0gt+Sin(Θ2)|1gt

Stufler et al

Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses

Cos(Θ2)|0gt+Sin(Θ2)|1gt

|1gt =gt electric charge

=gtPhotocurrent (PC)PC~Sin2(Θ2)

π-pulse corresponds to a population inversion

CNOT Dipole Coupling

Nearest neighbor interactions alter the energy states

Effective energy Ersquoi = Ei + sumjnei ∆Eij nj

Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied

Overcoming Short Interaction Distances

bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs

bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots

bull Optical Cavity coupling and fiber optical interconnects have also been proposed

Read Out of Specified Qubit States

bull Optical readoutExcitons decay spontaneously and the resulting radiation can be

measured

Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD

Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination

In either case repeated measurements have to be made A single shot readout still needs to be developed

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

• Quantum Dots
• Implementing Quantum Computers
• What is a quantum dot
• Gallium Arsenide Quantum Dots
• Energy Band Levels
• Slide 6
• Slide 7
• Tuning Quantum Dots
• Manufacturing methods
• Electron Beam Lithography
• Slide 11
• Molecular Beam Epitaxy
• Spin Quantum Computing
• Decoherence time ~ 100ns
• Artificial Atom
• Adding Electrons changing Vgate
• Comparison with Hydrogen
• Tuning the Quantum Dot
• Single Qubit manipulation
• Two Qubit Manipulation
• Ground State Splitting (J = Et ndash Es)
• Solving J(B(t)) Exchange Coupling
• SWAP - gate
• XOR ~ CNOT
• Read out Memory
• Another Spin up electron enters dot
• Spin down electron enters dot
• Slide 28
• 5 DiVincenzo QC Criteria
• The Physical System Excitons Trapped in GaAs Quantum Dots
• Initialization
• Relatively Long Decoherence Times
• A Universal set of Quantum Gates
• Single Qubit Gates Rabi Flopping
• Stufler et al
• CNOT Dipole Coupling
• Overcoming Short Interaction Distances
• Read Out of Specified Qubit States
• Slide 39

The Physical System Excitons Trapped in GaAs Quantum Dots

bull Exciton - a Coulomb correlated electron-hole pair in a semiconductor a quasiparticle of a solid

bull Often formed when photons excite electrons from the valence band into the conduction band

bull Wavefunctions are ldquohydrogen-likerdquo ie an ldquoexotic atomrdquo though the binding energy is much smaller and the extent much larger than hydrogen because of screening effects and the smaller effective masses

bull Decay by radiating photons Decay time ~50ps-1ns

bull Hence can define the computational basis as absence of an exciton |0gt or existence of an exciton |1gt

Initialization

bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to

prevent thermal excitations

bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the

register

Relatively Long Decoherence Times

bull Mechanismsndash Radiative Decay ~10ps-1ns

bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations

bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral

bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps

or less

A Universal set of Quantum Gates

bull Single Qubit Rotations through laser induced Rabi Oscillations

bull CNOT operations through dipole interactions and laser excitation

Single Qubit Gates Rabi Flopping

bull Light-particle interaction is characterized by the product of the dipole moment and the electric field

μbullE(t)= ħR(t)

Where R(t) is the Rabi frequency and the pulse area is given by

Θ(t)=intR(t)dt

and the state at time t is then given by

Cos(Θ2)|0gt+Sin(Θ2)|1gt

Stufler et al

Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses

Cos(Θ2)|0gt+Sin(Θ2)|1gt

|1gt =gt electric charge

=gtPhotocurrent (PC)PC~Sin2(Θ2)

π-pulse corresponds to a population inversion

CNOT Dipole Coupling

Nearest neighbor interactions alter the energy states

Effective energy Ersquoi = Ei + sumjnei ∆Eij nj

Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied

Overcoming Short Interaction Distances

bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs

bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots

bull Optical Cavity coupling and fiber optical interconnects have also been proposed

Read Out of Specified Qubit States

bull Optical readoutExcitons decay spontaneously and the resulting radiation can be

measured

Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD

Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination

In either case repeated measurements have to be made A single shot readout still needs to be developed

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

• Quantum Dots
• Implementing Quantum Computers
• What is a quantum dot
• Gallium Arsenide Quantum Dots
• Energy Band Levels
• Slide 6
• Slide 7
• Tuning Quantum Dots
• Manufacturing methods
• Electron Beam Lithography
• Slide 11
• Molecular Beam Epitaxy
• Spin Quantum Computing
• Decoherence time ~ 100ns
• Artificial Atom
• Adding Electrons changing Vgate
• Comparison with Hydrogen
• Tuning the Quantum Dot
• Single Qubit manipulation
• Two Qubit Manipulation
• Ground State Splitting (J = Et ndash Es)
• Solving J(B(t)) Exchange Coupling
• SWAP - gate
• XOR ~ CNOT
• Read out Memory
• Another Spin up electron enters dot
• Spin down electron enters dot
• Slide 28
• 5 DiVincenzo QC Criteria
• The Physical System Excitons Trapped in GaAs Quantum Dots
• Initialization
• Relatively Long Decoherence Times
• A Universal set of Quantum Gates
• Single Qubit Gates Rabi Flopping
• Stufler et al
• CNOT Dipole Coupling
• Overcoming Short Interaction Distances
• Read Out of Specified Qubit States
• Slide 39

Initialization

bull Register relaxes to the |00hellip0gt state within 50ps-1ns due to radiative decayndash Experimental systems are cooled to liquid helium temps ~4K to

prevent thermal excitations

bull Hence initialization with such a system is relatively easybull Other states can be initialized by applying gates to the

register

Relatively Long Decoherence Times

bull Mechanismsndash Radiative Decay ~10ps-1ns

bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations

bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral

bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps

or less

A Universal set of Quantum Gates

bull Single Qubit Rotations through laser induced Rabi Oscillations

bull CNOT operations through dipole interactions and laser excitation

Single Qubit Gates Rabi Flopping

bull Light-particle interaction is characterized by the product of the dipole moment and the electric field

μbullE(t)= ħR(t)

Where R(t) is the Rabi frequency and the pulse area is given by

Θ(t)=intR(t)dt

and the state at time t is then given by

Cos(Θ2)|0gt+Sin(Θ2)|1gt

Stufler et al

Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses

Cos(Θ2)|0gt+Sin(Θ2)|1gt

|1gt =gt electric charge

=gtPhotocurrent (PC)PC~Sin2(Θ2)

π-pulse corresponds to a population inversion

CNOT Dipole Coupling

Nearest neighbor interactions alter the energy states

Effective energy Ersquoi = Ei + sumjnei ∆Eij nj

Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied

Overcoming Short Interaction Distances

bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs

bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots

bull Optical Cavity coupling and fiber optical interconnects have also been proposed

Read Out of Specified Qubit States

bull Optical readoutExcitons decay spontaneously and the resulting radiation can be

measured

Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD

Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination

In either case repeated measurements have to be made A single shot readout still needs to be developed

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

• Quantum Dots
• Implementing Quantum Computers
• What is a quantum dot
• Gallium Arsenide Quantum Dots
• Energy Band Levels
• Slide 6
• Slide 7
• Tuning Quantum Dots
• Manufacturing methods
• Electron Beam Lithography
• Slide 11
• Molecular Beam Epitaxy
• Spin Quantum Computing
• Decoherence time ~ 100ns
• Artificial Atom
• Adding Electrons changing Vgate
• Comparison with Hydrogen
• Tuning the Quantum Dot
• Single Qubit manipulation
• Two Qubit Manipulation
• Ground State Splitting (J = Et ndash Es)
• Solving J(B(t)) Exchange Coupling
• SWAP - gate
• XOR ~ CNOT
• Read out Memory
• Another Spin up electron enters dot
• Spin down electron enters dot
• Slide 28
• 5 DiVincenzo QC Criteria
• The Physical System Excitons Trapped in GaAs Quantum Dots
• Initialization
• Relatively Long Decoherence Times
• A Universal set of Quantum Gates
• Single Qubit Gates Rabi Flopping
• Stufler et al
• CNOT Dipole Coupling
• Overcoming Short Interaction Distances
• Read Out of Specified Qubit States
• Slide 39

Relatively Long Decoherence Times

bull Mechanismsndash Radiative Decay ~10ps-1ns

bull Can be lengthened by electron-hole separationndash Background Electromagnetic fluctuations

bull Less of a problem than in other systems since the exciton and III-V heterostructure is on average electrically neutral

bull Gate times are determined by energy band spacing ie creation and annihilation energies ndash Gate operations for GaAs QDs are estimated at ~1ps

or less

A Universal set of Quantum Gates

bull Single Qubit Rotations through laser induced Rabi Oscillations

bull CNOT operations through dipole interactions and laser excitation

Single Qubit Gates Rabi Flopping

bull Light-particle interaction is characterized by the product of the dipole moment and the electric field

μbullE(t)= ħR(t)

Where R(t) is the Rabi frequency and the pulse area is given by

Θ(t)=intR(t)dt

and the state at time t is then given by

Cos(Θ2)|0gt+Sin(Θ2)|1gt

Stufler et al

Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses

Cos(Θ2)|0gt+Sin(Θ2)|1gt

|1gt =gt electric charge

=gtPhotocurrent (PC)PC~Sin2(Θ2)

π-pulse corresponds to a population inversion

CNOT Dipole Coupling

Nearest neighbor interactions alter the energy states

Effective energy Ersquoi = Ei + sumjnei ∆Eij nj

Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied

Overcoming Short Interaction Distances

bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs

bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots

bull Optical Cavity coupling and fiber optical interconnects have also been proposed

Read Out of Specified Qubit States

bull Optical readoutExcitons decay spontaneously and the resulting radiation can be

measured

Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD

Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination

In either case repeated measurements have to be made A single shot readout still needs to be developed

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

• Quantum Dots
• Implementing Quantum Computers
• What is a quantum dot
• Gallium Arsenide Quantum Dots
• Energy Band Levels
• Slide 6
• Slide 7
• Tuning Quantum Dots
• Manufacturing methods
• Electron Beam Lithography
• Slide 11
• Molecular Beam Epitaxy
• Spin Quantum Computing
• Decoherence time ~ 100ns
• Artificial Atom
• Adding Electrons changing Vgate
• Comparison with Hydrogen
• Tuning the Quantum Dot
• Single Qubit manipulation
• Two Qubit Manipulation
• Ground State Splitting (J = Et ndash Es)
• Solving J(B(t)) Exchange Coupling
• SWAP - gate
• XOR ~ CNOT
• Read out Memory
• Another Spin up electron enters dot
• Spin down electron enters dot
• Slide 28
• 5 DiVincenzo QC Criteria
• The Physical System Excitons Trapped in GaAs Quantum Dots
• Initialization
• Relatively Long Decoherence Times
• A Universal set of Quantum Gates
• Single Qubit Gates Rabi Flopping
• Stufler et al
• CNOT Dipole Coupling
• Overcoming Short Interaction Distances
• Read Out of Specified Qubit States
• Slide 39

A Universal set of Quantum Gates

bull Single Qubit Rotations through laser induced Rabi Oscillations

bull CNOT operations through dipole interactions and laser excitation

Single Qubit Gates Rabi Flopping

bull Light-particle interaction is characterized by the product of the dipole moment and the electric field

μbullE(t)= ħR(t)

Where R(t) is the Rabi frequency and the pulse area is given by

Θ(t)=intR(t)dt

and the state at time t is then given by

Cos(Θ2)|0gt+Sin(Θ2)|1gt

Stufler et al

Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses

Cos(Θ2)|0gt+Sin(Θ2)|1gt

|1gt =gt electric charge

=gtPhotocurrent (PC)PC~Sin2(Θ2)

π-pulse corresponds to a population inversion

CNOT Dipole Coupling

Nearest neighbor interactions alter the energy states

Effective energy Ersquoi = Ei + sumjnei ∆Eij nj

Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied

Overcoming Short Interaction Distances

bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs

bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots

bull Optical Cavity coupling and fiber optical interconnects have also been proposed

Read Out of Specified Qubit States

bull Optical readoutExcitons decay spontaneously and the resulting radiation can be

measured

Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD

Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination

In either case repeated measurements have to be made A single shot readout still needs to be developed

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

• Quantum Dots
• Implementing Quantum Computers
• What is a quantum dot
• Gallium Arsenide Quantum Dots
• Energy Band Levels
• Slide 6
• Slide 7
• Tuning Quantum Dots
• Manufacturing methods
• Electron Beam Lithography
• Slide 11
• Molecular Beam Epitaxy
• Spin Quantum Computing
• Decoherence time ~ 100ns
• Artificial Atom
• Adding Electrons changing Vgate
• Comparison with Hydrogen
• Tuning the Quantum Dot
• Single Qubit manipulation
• Two Qubit Manipulation
• Ground State Splitting (J = Et ndash Es)
• Solving J(B(t)) Exchange Coupling
• SWAP - gate
• XOR ~ CNOT
• Read out Memory
• Another Spin up electron enters dot
• Spin down electron enters dot
• Slide 28
• 5 DiVincenzo QC Criteria
• The Physical System Excitons Trapped in GaAs Quantum Dots
• Initialization
• Relatively Long Decoherence Times
• A Universal set of Quantum Gates
• Single Qubit Gates Rabi Flopping
• Stufler et al
• CNOT Dipole Coupling
• Overcoming Short Interaction Distances
• Read Out of Specified Qubit States
• Slide 39

Single Qubit Gates Rabi Flopping

bull Light-particle interaction is characterized by the product of the dipole moment and the electric field

μbullE(t)= ħR(t)

Where R(t) is the Rabi frequency and the pulse area is given by

Θ(t)=intR(t)dt

and the state at time t is then given by

Cos(Θ2)|0gt+Sin(Θ2)|1gt

Stufler et al

Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses

Cos(Θ2)|0gt+Sin(Θ2)|1gt

|1gt =gt electric charge

=gtPhotocurrent (PC)PC~Sin2(Θ2)

π-pulse corresponds to a population inversion

CNOT Dipole Coupling

Nearest neighbor interactions alter the energy states

Effective energy Ersquoi = Ei + sumjnei ∆Eij nj

Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied

Overcoming Short Interaction Distances

bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs

bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots

bull Optical Cavity coupling and fiber optical interconnects have also been proposed

Read Out of Specified Qubit States

bull Optical readoutExcitons decay spontaneously and the resulting radiation can be

measured

Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD

Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination

In either case repeated measurements have to be made A single shot readout still needs to be developed

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

• Quantum Dots
• Implementing Quantum Computers
• What is a quantum dot
• Gallium Arsenide Quantum Dots
• Energy Band Levels
• Slide 6
• Slide 7
• Tuning Quantum Dots
• Manufacturing methods
• Electron Beam Lithography
• Slide 11
• Molecular Beam Epitaxy
• Spin Quantum Computing
• Decoherence time ~ 100ns
• Artificial Atom
• Adding Electrons changing Vgate
• Comparison with Hydrogen
• Tuning the Quantum Dot
• Single Qubit manipulation
• Two Qubit Manipulation
• Ground State Splitting (J = Et ndash Es)
• Solving J(B(t)) Exchange Coupling
• SWAP - gate
• XOR ~ CNOT
• Read out Memory
• Another Spin up electron enters dot
• Spin down electron enters dot
• Slide 28
• 5 DiVincenzo QC Criteria
• The Physical System Excitons Trapped in GaAs Quantum Dots
• Initialization
• Relatively Long Decoherence Times
• A Universal set of Quantum Gates
• Single Qubit Gates Rabi Flopping
• Stufler et al
• CNOT Dipole Coupling
• Overcoming Short Interaction Distances
• Read Out of Specified Qubit States
• Slide 39

Stufler et al

Large wafer containing InGaAs QD was placed between a bias voltage and exposed to ultrafast laser pulses

Cos(Θ2)|0gt+Sin(Θ2)|1gt

|1gt =gt electric charge

=gtPhotocurrent (PC)PC~Sin2(Θ2)

π-pulse corresponds to a population inversion

CNOT Dipole Coupling

Nearest neighbor interactions alter the energy states

Effective energy Ersquoi = Ei + sumjnei ∆Eij nj

Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied

Overcoming Short Interaction Distances

bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs

bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots

bull Optical Cavity coupling and fiber optical interconnects have also been proposed

Read Out of Specified Qubit States

bull Optical readoutExcitons decay spontaneously and the resulting radiation can be

measured

Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD

Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination

In either case repeated measurements have to be made A single shot readout still needs to be developed

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

• Quantum Dots
• Implementing Quantum Computers
• What is a quantum dot
• Gallium Arsenide Quantum Dots
• Energy Band Levels
• Slide 6
• Slide 7
• Tuning Quantum Dots
• Manufacturing methods
• Electron Beam Lithography
• Slide 11
• Molecular Beam Epitaxy
• Spin Quantum Computing
• Decoherence time ~ 100ns
• Artificial Atom
• Adding Electrons changing Vgate
• Comparison with Hydrogen
• Tuning the Quantum Dot
• Single Qubit manipulation
• Two Qubit Manipulation
• Ground State Splitting (J = Et ndash Es)
• Solving J(B(t)) Exchange Coupling
• SWAP - gate
• XOR ~ CNOT
• Read out Memory
• Another Spin up electron enters dot
• Spin down electron enters dot
• Slide 28
• 5 DiVincenzo QC Criteria
• The Physical System Excitons Trapped in GaAs Quantum Dots
• Initialization
• Relatively Long Decoherence Times
• A Universal set of Quantum Gates
• Single Qubit Gates Rabi Flopping
• Stufler et al
• CNOT Dipole Coupling
• Overcoming Short Interaction Distances
• Read Out of Specified Qubit States
• Slide 39

CNOT Dipole Coupling

Nearest neighbor interactions alter the energy states

Effective energy Ersquoi = Ei + sumjnei ∆Eij nj

Hence a coherent π-pulse with energy Ersquot(nc) results in a state flop iff the control state is occupied

Overcoming Short Interaction Distances

bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs

bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots

bull Optical Cavity coupling and fiber optical interconnects have also been proposed

Read Out of Specified Qubit States

bull Optical readoutExcitons decay spontaneously and the resulting radiation can be

measured

Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD

Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination

In either case repeated measurements have to be made A single shot readout still needs to be developed

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

• Quantum Dots
• Implementing Quantum Computers
• What is a quantum dot
• Gallium Arsenide Quantum Dots
• Energy Band Levels
• Slide 6
• Slide 7
• Tuning Quantum Dots
• Manufacturing methods
• Electron Beam Lithography
• Slide 11
• Molecular Beam Epitaxy
• Spin Quantum Computing
• Decoherence time ~ 100ns
• Artificial Atom
• Adding Electrons changing Vgate
• Comparison with Hydrogen
• Tuning the Quantum Dot
• Single Qubit manipulation
• Two Qubit Manipulation
• Ground State Splitting (J = Et ndash Es)
• Solving J(B(t)) Exchange Coupling
• SWAP - gate
• XOR ~ CNOT
• Read out Memory
• Another Spin up electron enters dot
• Spin down electron enters dot
• Slide 28
• 5 DiVincenzo QC Criteria
• The Physical System Excitons Trapped in GaAs Quantum Dots
• Initialization
• Relatively Long Decoherence Times
• A Universal set of Quantum Gates
• Single Qubit Gates Rabi Flopping
• Stufler et al
• CNOT Dipole Coupling
• Overcoming Short Interaction Distances
• Read Out of Specified Qubit States
• Slide 39

Overcoming Short Interaction Distances

bull Electrostatic Dipole fields fall off as 1R^3 hence the CNOT gate can only be used for closely neighboring QDs

bull Solution Use a sequence of CNOTs on nearest neighbors to swap the desired qubits until they are contained in adjacent dots

bull Optical Cavity coupling and fiber optical interconnects have also been proposed

Read Out of Specified Qubit States

bull Optical readoutExcitons decay spontaneously and the resulting radiation can be

measured

Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD

Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination

In either case repeated measurements have to be made A single shot readout still needs to be developed

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

• Quantum Dots
• Implementing Quantum Computers
• What is a quantum dot
• Gallium Arsenide Quantum Dots
• Energy Band Levels
• Slide 6
• Slide 7
• Tuning Quantum Dots
• Manufacturing methods
• Electron Beam Lithography
• Slide 11
• Molecular Beam Epitaxy
• Spin Quantum Computing
• Decoherence time ~ 100ns
• Artificial Atom
• Adding Electrons changing Vgate
• Comparison with Hydrogen
• Tuning the Quantum Dot
• Single Qubit manipulation
• Two Qubit Manipulation
• Ground State Splitting (J = Et ndash Es)
• Solving J(B(t)) Exchange Coupling
• SWAP - gate
• XOR ~ CNOT
• Read out Memory
• Another Spin up electron enters dot
• Spin down electron enters dot
• Slide 28
• 5 DiVincenzo QC Criteria
• The Physical System Excitons Trapped in GaAs Quantum Dots
• Initialization
• Relatively Long Decoherence Times
• A Universal set of Quantum Gates
• Single Qubit Gates Rabi Flopping
• Stufler et al
• CNOT Dipole Coupling
• Overcoming Short Interaction Distances
• Read Out of Specified Qubit States
• Slide 39

Read Out of Specified Qubit States

bull Optical readoutExcitons decay spontaneously and the resulting radiation can be

measured

Alternatively an excitationprobe beam spot can be physically positioned in the region of the desired QD

Due to the statistical distribution of QD shape and size variations individual QDs can be more accurately identified and addressed through frequency discrimination

In either case repeated measurements have to be made A single shot readout still needs to be developed

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

• Quantum Dots
• Implementing Quantum Computers
• What is a quantum dot
• Gallium Arsenide Quantum Dots
• Energy Band Levels
• Slide 6
• Slide 7
• Tuning Quantum Dots
• Manufacturing methods
• Electron Beam Lithography
• Slide 11
• Molecular Beam Epitaxy
• Spin Quantum Computing
• Decoherence time ~ 100ns
• Artificial Atom
• Adding Electrons changing Vgate
• Comparison with Hydrogen
• Tuning the Quantum Dot
• Single Qubit manipulation
• Two Qubit Manipulation
• Ground State Splitting (J = Et ndash Es)
• Solving J(B(t)) Exchange Coupling
• SWAP - gate
• XOR ~ CNOT
• Read out Memory
• Another Spin up electron enters dot
• Spin down electron enters dot
• Slide 28
• 5 DiVincenzo QC Criteria
• The Physical System Excitons Trapped in GaAs Quantum Dots
• Initialization
• Relatively Long Decoherence Times
• A Universal set of Quantum Gates
• Single Qubit Gates Rabi Flopping
• Stufler et al
• CNOT Dipole Coupling
• Overcoming Short Interaction Distances
• Read Out of Specified Qubit States
• Slide 39

5 DiVincenzo QC Criteria

1 A scalable physical system with well-characterized qubits

2 The ability to initialize the state of the qubits to a simple fiducial state

3 Relatively long decoherence times compared to gate-operation times

4 A universal set of quantum gates

5 Qubit-specific measurement capability

• Quantum Dots
• Implementing Quantum Computers
• What is a quantum dot
• Gallium Arsenide Quantum Dots
• Energy Band Levels
• Slide 6
• Slide 7
• Tuning Quantum Dots
• Manufacturing methods
• Electron Beam Lithography
• Slide 11
• Molecular Beam Epitaxy
• Spin Quantum Computing
• Decoherence time ~ 100ns
• Artificial Atom
• Adding Electrons changing Vgate
• Comparison with Hydrogen
• Tuning the Quantum Dot
• Single Qubit manipulation
• Two Qubit Manipulation
• Ground State Splitting (J = Et ndash Es)
• Solving J(B(t)) Exchange Coupling
• SWAP - gate
• XOR ~ CNOT
• Read out Memory
• Another Spin up electron enters dot
• Spin down electron enters dot
• Slide 28
• 5 DiVincenzo QC Criteria
• The Physical System Excitons Trapped in GaAs Quantum Dots
• Initialization
• Relatively Long Decoherence Times
• A Universal set of Quantum Gates
• Single Qubit Gates Rabi Flopping
• Stufler et al
• CNOT Dipole Coupling
• Overcoming Short Interaction Distances
• Read Out of Specified Qubit States
• Slide 39