Hybrid quantum decoupling and error correction

Leonid Pryadko

University of California, Riverside

Pinaki Sengupta (LANL)Greg Quiroz (USC)Sasha Korotkov (UCR)

Yunfan Li (UCR)

Daniel Lidar (USC)

Outline

• Motivation: QEC and encoded dynamical decoupling with correlated noise

• General results on dynamical decoupling

• Concurrent application of logic

• Intercalated application of logic

• Conclusions and perspective

Stabilizer QECC

• Error correction is done by measuring the stabilizers frequently and correcting with the corresponding error operators if needed

• QECC period should be small compared to the decoherence rate

• Traditional QECCs: – Expensive: need many ancillas, fast measurement,

processing & correcting– May not work well with correlated environment

QECC with constant error terms

1 qubit

[[5,1,3]] [[5,1,5]]

[[3,1,3]]

QECC with constant error terms & decoupling

[[5,1,5]][[5,1,3]]

X Y

Q2

S1=X­X­I­I­I, S2=I­X­X­I­I,

S3=I­I­X­X­I, S4=I­I­I­X­X

[[5,1,5]]: fix

1- & 2-qubit

phase errors

1-qubit symmetric seq.

Combined coherence protection technique

• Passive: Dynamical Decoupling– Effective with low-frequency bath– Most frugal with ancilla qubits needed– Needs fast pulsing (resource used: bandwidth)

• Active: Quantum error correcting codes– Most universal– Needs many ancilla qubits– Needs fast measurement, processing & correcting– Expensive

• Combined: Encoded Dynamical Recoupling [Viola, Lloyd & Knill (1999)] – Better suppression of decoherence due to slow

environment potentially much more efficient– Control can be done along with decoupling

• Errors are fully reversed at the end of the decoupling cycle

• Normalizer and stabilizer commute – add logic anywhere!?

Example with hard pulses & constant errors

1 2

XL YL ZL1 2

• Errors are fully reversed at the end of the decoupling cycle

• Normalizer and stabilizer commute – add logic anywhere!

Example with hard pulses & constant errors

1 2

XL YL ZL1 2

XL YL ZL

1 2

Error operators in rotating frame

• S: system, E: environment, DD: dynamical decoupling

• Dynamical decoupling is dominant: is large • Solve controlled dynamics and write the Hamiltonian in the

interaction representation with respect to DD

• Interaction representation with respect to environment

• Bath coupling is now modulated at the combination of the environment and dynamical decoupling frequencies

• With first-order average Hamiltonian suppressed, all S+E coupling is shifted to high frequences no T1 processes

(Kofman & Kurizki, 2001)

Resonance shift with decoupling

F()

|0i |1i~

system spectrum

Environment spectrum

• Slowly-evolving system couple strongly to low- noise

• Decoupling with period 2/ suppresses the low- spectral peak & creates new peaks shifted by n

• Noise decoupling similar with lock-in techniques

with refocusing

Resonance shift with decoupling

F()

|0i |1i~

system spectrum

Environment spectrum

• Slowly-evolving system couples strongly to low- noise

• Decoupling with period 2/ suppresses the low- spectral peak & creates new peaks shifted by n

• Noise decoupling similar with lock-in techniques

with refocusing

By analyticity, reactive processes should also be affected

Quantum kinetics with DD: results

• K=0 (no DD): Dephasing rate » max(J,(0)0),

(t)=||hB(t)B(0)i||• K=1 (1st order): Single-phonon decay eliminated

Dephasing rate » maxJ2,(0), plus effect of higher order derivatives of (t) at t=0.

Reduction by factor

• K=2 (2nd order): all derivatives disappear

Exponential reduction in

• Visibility reduction »(0)2 (generic sequence)

»’’(0)»(0) (symmetric sequence)

(LPP & P. Sengupta, 2006)

Encoded dynamical recoupling

• Several physical qubits logical• Operators from the stabilizer are used for

dynamical decoupling ( ), at the same time running logic operators from

• It is important that mutually commute

(Viola, Lloyd & Knill,1999)

No-resonance condition for T1 processes

mutually commute• Interaction representation

• Combination of three rotation frequencies– Harmonics of DD (periodic)

– L (can be small since logic is not periodic)

– E (limited from above by Emax)

• State decay through environment is suppressed if

No-resonance: spectral representation

F()

system spectrum

Environment

spectral function

• DD pulses shift the system’s spectral weight to higher frequencies

• Simultaneous execution of non-periodical algorithm widens the corresponding peaks

• More stringent condition to avoid the overlap with the spectrum of the environmental modes

with refocusing

with DD & Logic

Recoupling with concurrent logic

4-pulse

XL YL ZL

1 2

Recoupling with concurrent logic: expand

4-pulse L=4

XL YL ZL

1 2

Intercalated pulse application • Apply logical pulses at the

end of the decoupling interval– With hard pulses, this cancels

the average error over decoupling period [Viola et al, 1999]

– Overlap with bath is power-law in c

– Equivalently, visibility reduction with each logic pulse

– With finite-length pulses, additional error depending on pulse duration and precise placement

• Use shaped pulses to construct sequences with no errors to 1st or 2nd order

F()

system spectrum

Environment

spectral function

with refocusing

with DD & Logic

Power of

Recoupling with intercalated logic

1 2 XL YL ZL

4-pulse

Recoupling with intercalated logic (cont’d)

1 2 XL YL ZL

4-pulse

Compare at t/p=384

Intercalated

Concurrent Concurrent

Concurrent

Conclusions and Outlook

• Much mileage can be gained from carefully engineered concatenation– With decoupling at the lowest level, need careful pulse

placement, pulse & sequence design

• Bandwidth is used to combine logic and decoupling

• Still to confirm predicted parameter scaling• Analyze effects of:

– Actual many-qubit gates needed– Fast decoherence addition– QEC dynamics (gates with ancillas, measurement,…)

• Can fault-tolerance be achieved in this scheme?