Superconducting quantum bitsnanowiki.phy.bme.hu/images/5/53/Qubit_part2.pdfY. Nakamura et al.,...

Superconducting quantum bits

Péter Makk

BUTE, Low temperature solid state physics laboratoryQubits

Qubit = quantum mechanical two level system

DiVincenzo criteria for quantum computation:

1. Register of 2-level systems (qubits), n = 2N states: eg. |101..01> (N qubits)2. Initialization: e.g. setting it to |000..00>3. Tools for manipulation: 1- and 2-qubit gates, e.g. Hadamard gates: UH|0> = (|0> + |1>)/2,and CNOT gates to create entangled states, UCNOTUH|00 >=(|00> + |11>)/2 4. Read-out : |ψ> = a|0> + bei|1> → a, b5. Long decoherence times: > 104 2-qubit gate operations needed for error correction to maintain coherence ”forever”.6. Transport qubits and to transfer entanglement between different coherent systems (quantum-quantum interfaces). 7. Create classical-quantum interfaces for control, readout and information storage.

Two level system (quasi-spin)

basis, general state by rotation:

Representation : Bloch sphere

BUTE, Low temperature solid state physics laboratoryQubits

All operations are possible with one-qubit and one two-qubit gatese.g. CNOT: first control bit, second not

Gates: Unitary operators

N-qubit states

BUTE, Low temperature solid state physics laboratoryJosephson junctions

Josephson equationsS1 S2I

I

eV

2

-2

Voltage biased junction:

eV2

()

Applying a constant bias voltage:

An AC current with =2eV/ħ is flowing. The DC current averages to zero.


Equation of motion

potential

I

„washboard potential”

S1 S2I

plasma oscillationsfor I>Ic DC voltage appears

U


S1 S2I

Current biased junction:Switching at the critical current!

I

eV2

-2

I

eV

I

eV2

-2

overdampedsmall R and C

underdampedlarge R and C

keeps sliding due to kinetic energy


Josephson junctions is a non-linear inductancethe energy spectra is anharmonic

for small

resonant circuit

qubit is separateable

Flux quantization phase difference flux

BUTE, Low temperature solid state physics laboratoryrf-squid

BA

1 Equation of motion

potential

integer bias flux

half-integer bias flux

For half integer quantum, two minima:two persistent current states, circulating in different direction

U

BUTE, Low temperature solid state physics laboratorydc-squid

BA

1

2

/cossin2/2sin/2sin 00021 eIIIII cc

If the critical current of the two qubits are the sameThe maximal value of the critical current is tuned by the magnetic flux: /cos2max eII c

Dependence of the maximum supercurrent (critical current) through the two-junction interferometer on the total magnetic flux through its interior.

The strong dependence of Icon flux makes the DC-SQUID an extremely sensitive flux detector.

BUTE, Low temperature solid state physics laboratoryCooper pair box

Jospehson energy:

Electrostatic energy (like for a Qdot):

For two josephson junctions:

The phase difference can be tuned by applying a magnetic field. Thus the Josephson energy can be tuned

normal CPB

split CPB

BUTE, Low temperature solid state physics laboratorySecond quantization

Commutation relation

Phase and number are conjugated variables!

Phase representation (analogue: coordinate representation)

N operator and N eigenstates: and

in the other representation

transfer of Cooper pairs

BUTE, Low temperature solid state physics laboratoryCharge qubit

0-2 2

For EJ << EC

The charging energy states are degenerate at the crossing point: EJ makes a transition and lifts the degeneracy:

The charge qubit near the degeneracy point

where :

At the degeneracy point the eigenstates are trivial:

Far from the degeneracy points the original states are good eigenstates.

Higher order coupling through EJ are negligible

BUTE, Low temperature solid state physics laboratoryCharge qbit

Y. Nakamura et al., Nature, 398, 786 (1999)

• First the qubit is prepared in state |0>• Fast DC pulse to the gate not adiabatic, it remains in |0>• It starts Rabi-oscillating, and evolves during the pulse length (t)

• If after the pulse, the qubit is in |1> it will relax back and the current is detected (high repetition frequency) – single shot readout• Detection: working point for state 1 is above delta in energydecay to quasi particles

• By adjusting t, the length of the pulse Rabi oscillation is seen• DC gate is need to calibrate the pulse gate• Relaxation <10ns, probably due to charge fluctuations

Cooper pairnumber

BUTE, Low temperature solid state physics laboratoryRabi oscillations

Other readout for charge qubit: with SET

BUTE, Low temperature solid state physics laboratoryRelaxation - T1 and T2

Adding a small modulating field

In rotating frame with angular moment (0,0,) , H1 is static: H1’=(H1, 0 , 0)

If L, than the magnetic moment will precess around H1’With short pulses, the moment can be rotated with arbitrary angleE.g. /2 pulse rotates to y’ axis, a precession in the x-y plane in the lab frame

Larmor precession

where the larmor frequency is:

BUTE, Low temperature solid state physics laboratoryRelaxation - T1 and T2

T1 (spin-lattice in NMR)

T2 – decoherence time (spin-spin in NMR)

T1

The time until a superposition preserves its phaseIn NMR lot of spins, which precess with different larmor frequency, and the net spin disappears

Relaxation – Bloch equations

BUTE, Low temperature solid state physics laboratoryQuantronium

-charge qubits operated at the degeneracy point are not sensitive to charge fluctuations (2nd order) but to phase fluctuations- for charge-flux qubit there is also a sweet spot in the flux-charge plain (Quantronium)

Quantronium is operated at the sweet point, where Ib=0. Except the sweet point a net supercurrent flows, the direction depending on the stateAnother, more transparent Josephson junction, with large EJ is addedFor readout Ib is applied such, that Ib+I0<Ic, but Ib+I1>Ic, and finite voltage is measured

D. Vion et al., Science 296 (2002)


Rabi oscillations Ramsey oscillations

after free precession, /2 rotates up or down, depending on the acquired phaseT2 follows from the decay of the oscillation

Rotation for time, drive between the two states after the rotation measurementDecay of oscillations go with TRabiFrequency depends on the microwave power

D. Vion et al., Science 296 (2002)


Measurement of T1

- T1 is the longest scale and it has also contribution to T2-With echo technique stationary inhomogenities can be filtered out- from the resonance signal ~ T2 can be deduced

Spin-echo

D. Vion et al., Science 296 (2002)Grégoire Ithier PhD Thesis

BUTE, Low temperature solid state physics laboratoryTransmon

Problem with charge qubit: sensitivity to charge noise, fluctuations of the gate electrodeThe degeneracy point is a sweet point, sensitivity to charge fluctuations is second order

By increasing the Ej/Ec ratio:

- good news: the dispersion of the bands disappear exponentially

- bad news: the anharmonicity decreases (even changes sign), but just linearly, still enough to separate qubit

J. Koch et al., PRA 76, 042319 (2007)


Ej/ES increased by a shunting capacitance (EC decreased): design parameter

T1 ~ sT ~10s


Coupling to resonator

Coupling:

d and E higher than in 3d cavityeasy to arrive at strong coupling regime

RWAJaynes-Cummings Hamiltonian

without coupling(g=0)

A. Blais et al., PRA 69, 062320 (2004)


Resonant case, g≠0

Vacuum Rabi splitting can be seen by measuring the transmission of the cavity

A. Walraff et al., Nature 431, 9 (2004)


Non resonant case, g≠0dispersive regime: r-a>> g

- State dependent polarizability of ‘atom’ pulls the cavity frequency

Lamb shiftqubit ac Stark shift and cavity shift

Example: spectroscopy of CPB charge qubit

The transition energies mapped with sophisticated setup…

D. Schuster et al., PRL 94 (2005), 123602.

BUTE, Low temperature solid state physics laboratoryPhase qubit

Operation:- Prepare eg. to state 1Readout:- change Ic, such that 1 can tunnel out finite voltage appears on the junction- or swap 1-2, and 2 can tunnel out

Rabi oscillations etc…

Tune the potential of a sing JJ (washboard potential) , such, that it is asymmetric and only houses 2-3 levelsEc<<EJ, I<Ic

Tilt the potential such, that it only houses 1 or 2 levels

a bit different design (flux controlled phase qubit)

flux biasdc squid readout

Problems: always connected to readout , quasiparticles shunt the junction, long recondensation time…


Coupled phase qubitsCapacitive coupling

- The levels (splitting) of the single qubits are tunable- If the level spacing is nearly the same than the states of the two qubits hybridize- First three levels |00>, |01> - |10> and |01> + |10>

Escape probability is measured

Ib1 is fix, f is fix and Ib2 is tunedFor fix f, Ib2 will tune the coupled levels, and for some Ib2 the condition is fulfilled

On the f--Ib2 map avoided crossing is seencoupling

R. McDermott et al., Science 307 (2005)


Readout of both qubits at the same timeQbits are brought to resonanceAt resonance |01> and |10> are not eigenstatesPrepare |00>, than excite qbit1 |10>This is not an eigenstate and start to precessMeasurement of the two qubits is anticorrelatedCoherence time is the same as for single qubit

At t=hbar/2S, |10> goes to i| 01> iSwap

R. McDermott et al., Science 307 (2005)

BUTE, Low temperature solid state physics laboratoryState tomography

i

iiip ||

Density matrix

If the whole system is entangled with the environment, than the reduced matrix is not pureDecoherence: offdiagonal elements

For a single qubit

zyxkkkr

,,,01110

0100

21

and

1100

1001

1001

11000

)(

1

z

y

x

rir

rr

)1(21)(

21|)00|( 00 zz rTrP

By projective measurement (one Stern-Gerlach measurement)

Other components, eg. rx comes from the rotation around y by /2 so, that x and z are interchanged.More precisely by rotating with t angle and measuring fitting the angle dependence, rx is obtained.


Coupled phase qubitsAt resonance: oscillation with S/h=10MHZ

freq between |01> and i|10> 2-qubit operation

|00> -> |01> not eigenstate, starts to evolve

10|)2

sin(01|)2

cos()(|

StiStt

- tfree=25 ns: entangled state:

But: (pulse length not negligible) - after tfree=16 ns 90z rotation:

eigenstate, does not show any oscillationsTo show it is not the destruction of coherence:-> 180z pulse, and starts oscillating again

)10|01(|2

1| 1 i

)10|01(|2

1|

)10|01(|2

1| i

10int |)0110||1001(|2

CCSwhereSH x


)10|01(|2

1| 1 i

State tomography on the state:

Single qubit fidelities:F0=0.95, F1=0.85

Fidelity for |ψ1> F=0.75After correction with single qubit fidelities: F=0.87

Estimated maximal fidelity: F=0.89

Cause of fidelity loss:• single qubit decoherence

BUTE, Low temperature solid state physics laboratoryFlux qbit

Two wells two levels – for symmetric potential degenerate flux statesThe two states have opposite persistent currentIf tunneling in switched on (), states hybridize and the macroscopic tunneling determines the separation

Hard to fabricate, big loop is needed for inductance matching (big decoherence) 3 JJ-s qbit

where

potential for half integer flux bias

3-junction flux qubit

the potential is parabolic on the white intersectionI. Chiorescu et al., Science 299 (2003)

barrier height

assymetry by flux

BUTE, Low temperature solid state physics laboratoryFlux qubit

The current as a function of the fluxAway from half flux quanta, pure flux states

Detection – by DC squid measuring the opposite supercurrents

A small change in the squid signalshows the persistent current states

Caspar H. van der Wal et al., Science 290 (2000)

BUTE, Low temperature solid state physics laboratoryFlux qbit

Ibias

During the sweeping of the magnetic field, microwave applied- the resonance seen for different frequencies at different flux points- peaks indicate switching between flux states- the spectra is nicely reproduced

Caspar H. van der Wal et al., Science 290 (2000)

BUTE, Low temperature solid state physics laboratoryC-NOT with coupled flux qubits

If control qubit is 1, it flips the target qubithere, the first qubit is the control qubit, |10>|11> and |11>|10>

Qubits are operated far from the degeneracy pointMain idea: If one of the qubits manipulated, e.g.. qubit 1 is excited, the change of the persistent current will change the resonance frequency of qubit 2Qubit1= Control, Qubit2=Targetstates 0C0T, 0C1T, 1C0T, 1C1T Using microwave pulses for rotations (its phase determines the axes)

With this rotation qubit 1 controlled CNOT is realized (almost, but phase can be easily shifted)

J. H. Plantenberg et al., NATURE 447 (2007)

BUTE, Low temperature solid state physics laboratoryC-NOT with coupled flux qubits

- preparation of state- CNOT- measurement of qubit states

e.g. first rotation prepares from |00> a|00>+ b|10>than the CNOT is only resonant for |10>|10>|11>|00>|00>With other preparation sequence other 2 relationsFrom readout P00 and P10 … can be determined

control target

11

00

10

00

10

Target result

J. H. Plantenberg et al., NATURE 447 (2007)

BUTE, Low temperature solid state physics laboratorySummary

Superconducting Circuits and Quantum InformationJ. Q. You and Franco NoriPhys. Today 58(11), 42 (2005)

Superconducting Quantum Circuits, Qubits and ComputingG. Wendin and V.S. ShumeikoarXiv:cond-mat/0508729v1

BUTE, Low temperature solid state physics laboratorySummary

Superconducting quantum bitsnanowiki.phy.bme.hu/images/5/53/Qubit_part2.pdfY. Nakamura et al.,...

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