Hybrid quantum decoupling and error correction
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Transcript of Hybrid quantum decoupling and error correction
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?