Semiconductor Qubits for Quantum Computation
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Transcript of Semiconductor Qubits for Quantum Computation
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Semiconductor qubits forquantum computation
Matthias FehrTU Munich
JASS 2005
St.Petersburg/Russia
Is it possible to realize a quantum
computer with semiconductor technology?
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Contents
1. History of quantum computation
2. Basics of quantum computationi. Qubits
ii. Quantum gates
iii. Quantum algorithms
3. Requirements for realizing a quantum computer
4. Proposals for semiconductor implementationi. Kane concept: Si:31Pii. Si-Ge heterostructure
iii. Quantum Dots (2D-electron gas, self-assembly)
iv. NV-
center in diamond
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The history of quantum computation
1936 Alan Turing
1982 Feynman
1985 Deutsch
Church-Turing thesis:
There is a Universal Turing machine, that can efficiently
simulate any other algorithm on any physical device
Computer based on quantum mechanics mightavoid problems in simulating quantum mech.systems
Search for a computational device to simulate anarbitrary physical system
quantum mechanics -> quantum computer
Efficient solution of algorithms on a quantum computerwith no efficient solution on a Turing machine?
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1994 Peter Shor
1995 Lov Grover
In the 1990s
1995 Schumacher
1996 Calderbank,Shor, Steane
Efficient quantum algorithms
- prime factorization
- discrete logarithm problem->more power
Efficient quantum search algorithm
Efficient simulation of quantum mechanical systems
Quantum bit or qubit as physical resource
Quantum error correction codes
- protecting quantum states against noise
The history ofquantum computation
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The basics of quantum computation
Classical bit: 0 or 1
2 possible values
Qubit:
are complex -> infinite possiblevalues -> continuum of states
10 +=
,
Qubit measurement: result 0 with probability
result 1 with probability
Wave function collapses during measurement,
qubit will remain in the measured state
2
2
122
=+
0010 %1002
+=
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Qubits
Bloch sphere:
We can rewrite our state with
phase factors
Qubit realizations: 2 level systems
1) ground- and excited states of electron orbits
2) photon polarizations
3) alignment of nuclear spin in magnetic field4) electron spin
+= 1
2sin0
2cos
iiee
,,
Bloch sphere [from Nielson&Chuang]
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Single qubit gates
Qubits are a possibility to storeinformation quantum mechanically
Now we need operations toperform calculations with qubits
-> quantum gates:
NOT gate:classical NOT gate: 0 -> 1; 1 -> 0
quantum NOT gate:
Linear mapping -> matrixoperation
Equal to the Pauli spin-matrix
0110 ++
xX =
01
10
=
+
X
10
X
X
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Single qubit gates
Every single qubit operation can bewritten as a matrix U
Due to the normalization condition everygate operation U has to be unitary
-> Every unitary matrix specifies a valid
quantum gate
Only 1 classical gate on 1 bit, but
quantum gates on 1 qubit.
Z-Gate leaves unchanged, and flipsthe sign of
Hadamard gate = square root of NOT
IUU =*
=
=
11
11
2
1
10
01
H
Z
0
11
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Hadamard gate
Bloch sphere:
- Rotation about the y-axis by 90
- Reflection through the x-y-plane
Creating a superposition
2
10
2
1010
+
++ H
Bloch sphere [from Nielson&Chuang]
2101
2
100
+
H
H
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Decomposing single qubit operations
An arbitrary unitary matrix can bedecomposed as a product of rotations
1st
and 3rd
matrix: rotations about the z-axis
2nd matrix: normal rotation
Arbitrary single qubit operations with a finiteset of quantum gates
Universal gates
=
2/
2/
2/
2/
0
0
2/cos2/sin
2/sin2/cos
0
0
i
i
i
i
i
e
e
e
eeU
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Multiple qubits
For quantum computation multiple qubits are needed!
2 qubit system:
computational bases stats:superposition:
Measuring a subset of the qubits:
Measurement of the 1st qubit gives 0 with probability
leaving the state
11,10,01,00
11100100 11100100 +++=
1st qubit 2nd qubit
2
01
2
00 +
2
01
2
00
0100 0100`
+
+=
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Entanglement
Bell state or EPR pair:
prepare a state:
Measuring the 1st qubit gives
0 with prop. 50% leaving
1 with prop. 50% leaving
The measurement of the 2nd qubits alwaysgives the same result as the first qubit!
The measurement outcomes are correlated!
Non-locality of quantum mechanics
Entanglement means that state can not bewritten as a product state
11=
2
1100 +=
00=
2
1100
2
11011000
2
10
2
1021
+
+++
=
+
+==
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Multiple qubit gates, CNOT
Classical: AND, OR, XOR, NAND, NOR -> NAND is universal
Quantum gates: NOT, CNOT
CNOT gate:
- controlled NOT gate = classical XOR
- If the control qubit is set to 0, target qubit is the same
- If the control qubit is set to 1, target qubit is flipped
CNOT is universalfor quantum computation
Any multiple qubit logic gate may be composed from CNOT and single qubit gates
Unitary operations are reversible
(unitary matrices are invertible, unitary -> too )
Quantum gate are always reversible, classical gates are not reversible
2mod,,
1011;1110;0101;0000
ABABA
=
11
10
01
00
0100
10000010
0001
CNU
U 1U
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Qubit copying?
classical: CNOT copies bits
Quantum mech.: impossible
We try to copy an unknown state
Target qubit:
Full state:
Application of CNOT gate:
We have successful copied , but only in the case
General state
No-cloningtheorem: major difference between quantum andclassical information
10 ba +=
0
[ ] 1000010 baba +=+
=+ 1100 ba
10 or=
11100100
22
bababa+++=
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Quantum parallelism
Evaluation of a function:
Unitary map: black box
Resulting state:
Information on f(0) and f(1) witha single operation
Not immediately useful, becauseduring measurement thesuperposition will collapse
}1,0{}1,0{:)( xf
)(,,: xfyxyxUf
2)1(,1)0(,0 ff
+=
)1(,1
)0(,0
f
f
Quantum gate [from Nielson&Chuang (2)]
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Deutsch algorithm
Input state:
Application of :
010 =
( ) 2/10210
2
101 =
+
= x
( ) 2/10)1( )(2 = xxf
fU
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Deutsch algorithm
)1()0(1)1()0(
)1()0(0)1()0(
ffifff
ffifff
===
=
2
10)1()0(3 ff
global property determined withone evaluation of f(x)
classically: 2 evaluations needed
Faster than any classical device
Classically 2 alternatives excludeone another
In quantum mech.: interference
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Quantum algorithms
22 )(log nN =n
n
N
nNN
2
2)log(
=
=
Classical steps quantum logic steps
Fourier transform
e.g.:- Shors prime factorization
- discrete logarithm problem
- Deutsch Jozsa algorithm
- n qubits
- N numbers
- hidden information!
- Wave function collapseprevents us from directlyaccessing the information
Search algorithms
Quantum simulation cN bits kn qubits
N N
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The Five Commandments of DiVincenzo
1. A physical system containing qubits is needed
2. The ability to initialize the qubit state
3. Long decoherence times, longer than the gate
operation time Decoherence time: 104-105 x clock time
Then error-correction can be successful
4. A universal set of quantum gates (CNOT)
5. Qubit read-out measurement
...000
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Realization of a quantum computer
Systems have to be almostcompletely isolated from their
environment
The coherent quantum state hasto be preserved
Completely preventingdecoherence is impossible
Due to the discovery of quantumerror-correcting codes, slightdecoherence is tolerable
Decoherence times have to bevery long -> implementation
realizable
Performing operations on severalqubits in parallel
2- Level system as qubit:
Spin particles
Nuclear spins
Read-out:
Measuring the single spin states
Bulk spin resonance
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Si:31P, Kane concept from 1998
Logical operations on nuclear spinsof31P(I=1/2) donors in a Si host(I=0)
Weakly bound 31P valence electron atT=100mK
Spin degeneracy is broken by B-field Electrons will only occupy the lowest
energy state when
Spin polarization by a strong B-field
and low temperature
Long 31P spin relaxation time ,
due to low T
s1810
TkBB >>2
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Single spin rotations
Hyperfine interaction at the nucleus
: frequency separation of the nuclear levels
A-gate voltage pulls the electron wave functionenvelope away from the donors
Precession frequencyof nuclear spins iscontrollable
2nd magnetic field Bac in resonance to thechanged precession frequency
Selectively addressing qubits
Arbitrary spin rotations on each nuclear spin
2A
B
AABgh
ABgBH
B
nnA
nen
znn
e
zBen
2222 ++=
+=
A
h2/acBeff Bg
=
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Qubit coupling
J-gates influence the neighboringelectrons -> qubit coupling
Strength of exchange couplingdepends on the overlap of thewave function
Donor separation: 100-200
Electrons mediate nuclear spininteractions, and facilitatemeasurement of nuclear spins
o
A
+++=
BBB
eenen
a
r
a
r
a
erJ
JAABHH
2exp6.1)(4
)(2/5
2
1222211
e2
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Qubit measurement
J < BB/2: qubit operation
J > BB/2: qubit measurement
Orientation of nuclear spin 1 alonedetermines if the system evolvesinto singlet or triplet state
Both electrons bound to samedonor (D- state, singlet)
Charge motion between donors
Single-electron capacitancemeasurement
Particles are indistinguishable
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Many problems
Materials free of spin(isotopes)
Ordered 1D or 2D-donor array
Single atom doping methodes
Grow high-quality Si layers onarray surface
100-A-scale gate devices
Every transistor is individual ->large scale calibration
A-gate voltage increases theelectron-tunneling probability
Problems with low temperatureenvironment
Dissipation through gate biasing
Eddy currents by Bac
Spins not fully polarized
0I
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SRT with Si-Ge heterostructures
Spin resonance transistors, at a size of2000 A
Larger Bohr radius (larger )
Done by electron beam lithography
Electron spin as qubit
Isotropic purity not critical
No needed spin transfer betweennucleus and electrons
Different g-factors
Si: g=1.998 / Ge: g=1.563 Spin Zeeman energy changes
Gate bias pulls wave function awayfrom donor
*,m
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Confinement and spin rotations
Confinement through B-layer
RF-field in resonance with SRT -> arbitrary spin phase change
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2-qubit interaction
No J-gate needed
Both wave functions are pullednear the B-layer
Coulomb potential weakens
Larger Bohr radius
Overlap can be tuned
CNOT gate
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Detection of spin resonance
FET channel:
n-Si0.4Ge0.6 ground plane counter-electrode
Qubit between FET channel andgate electrode
Channel current is sensitive to
donor charge states: ionized / neutral /
doubly occupied (D- state)
D- state (D+ state) on neighbortransistors, change in channelcurrent -> Singlet state
Channel current constant -> tripletstate
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Electro-statically defined QD
GaAs/AlGaAsheterostructure -> 2DEG
address qubits with
high-g layer
gradient B-field
Qubit coupling by loweringthe tunnel barrier
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Single spin read-out in QD
Spin-to-charge conversion of electronconfined in QD (circle)
Magnetic field to split states
GaAs/AlGaAs heterostructure -> 2DEG
Dots defined by gates M, R, T
Potential minimum at the center
Electron will leave when spin-
Electron will stay when spin-
QPC as charge detector
Electron tunneling between reservoirand dot
Changes in QQPC detected bymeasuring IQPC
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Two-level pulse on P-gate
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Self-assembled QD-molecule
Coupled InAs quantum dots quantum molecule
Vertical electric field localizescarriers
Upper dot = index 0
Lower dot = index 1
Optical created exciton
Electric field off -> tunneling ->entangled state
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Self-assembled Quantum Dots array
Single QD layer
Optical resonant excitation of e-hpairs
Electric field forces the holes intothe GaAs buffer
Single electrons in the QD ground
state (remains for hours, at low T)
Vread: holes drift back andrecombine
Large B-field: Zeeman splitting ofexciton levels
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Self-assembled Quantum Dots array
Circularly polarized photons
Mixed states
Zeeman splitting yields either
Optical selection of pure spinstates
=
+=
eJ
eJ
ze
ze
h
h
2/1
2/1
,
,
=
+=
hJ
hJ
zh
zh
h
h
2/3
2/3
,
,
=+=
+
+
heandhe
JJJ zhzez h
h
h
1
1
1
,,
heorhe
h
h
1
1
+=
=
Jhe
Jhe
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NV- center in diamond
Nitrogen Vacancy center: defect indiamond, N-impurity
3A -> 3E transition: spin conserving
3E -> 1A transition: spin flip
Spin polarization of the ground state Axial symmetry -> ground state
splitting at zero field
B-field for Zeeman splitting of tripletground state
Low temperature spectroscopy
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NV- center in diamond
Fluorescence excitation with laser
Ground state energy splittinggreater than transition line with
Excitation line marks spinconfiguration of defect center
On resonant excitation:
Excitation-emission cycles 3A -> 3E
bright intervals, bursts
Crossing to 1A singlet small
No resonance
Dark intervals in fluorescence
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Thank you very much!
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References
1. Kane, B.E. A silicon-based nuclear spin quantum computer, Nature 393, 133-137 (1998)
2. Nielson & Chuang, Quantum Computation and Quantum Information, Cambridge University Press (2000)
3. Loss, D. et al. Spintronics and Quantum Dots, Fortschr. Phys. 48, 965-986, (2000)
4. Knouwenhoven, L.P. Single-shot read-out of an individual electron spin in a quantum dot, Nature 430, 431-435(2004)
5. Forchel, A. et al., Coupling and Entangling of Quantum States in Quantum Dot Molecules, Science 291, 451-453(2001)
6. Finley, J. J. et al., Optically programmable electron spin memory using semiconductor quantum dots, Nature432, 81-84 (2004)
7. Wrachtrup, J. et al., Single spin states in a defect center resolved by optical spectroscopy, Appl. Phys. Lett. 81(2002)
8. Doering, P. J. et al. Single-Qubit Operations with the Nitrogen-Vacancy Center in Diamond, phys. stat. sol. (b)233, No. 3, 416-426 (2002)
9. DiVincenzo, D. et al., Electron Spin Resonance Transistors for Quantum Computing in Silicon-GermaniumHeterostructures, arXiv:quant-ph/9905096 (1999)
10. DiVincenzo, D. P., The Physical Implementation of Quantum Computation, Fortschr. Phys. 48 (2000) 9-11, 771-783