hybrid quantum decoupling and error correction
DESCRIPTION
Hybrid quantum decoupling and error correction. University of California, Riverside. Leonid Pryadko. Yunfan Li (UCR) Daniel Lidar (USC). Pinaki Sengupta( LANL ) Greg Quiroz (USC) Sasha Korotkov (UCR). Outline. - PowerPoint PPT PresentationTRANSCRIPT
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?