Thresholds for Universal Concatenated Quantum Codes

Chamberland, Christopher; Jochym-O’Connor, Tomas; Laflamme, Raymond

doi:10.1103/physrevlett.117.010501

Cited by 41 publications

(63 citation statements)

References 22 publications

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“…Only qubits within each code block undergo correlated noise described by eq. (23). The noise dynamics for each code block of the code C out will thus be described by the effective noise dynamics of eq.…”

Section: E Effective Noise Channels For Concatenated Codesmentioning

confidence: 99%

Hard decoding algorithm for optimizing thresholds under general Markovian noise

Chamberland

Wallman

et al. 2017

Phys. Rev. A

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Quantum error correction is instrumental in protecting quantum systems from noise in quantum computing and communication settings. Pauli channels can be efficiently simulated and threshold values for Pauli error rates under a variety of error-correcting codes have been obtained. However, realistic quantum systems can undergo noise processes that differ significantly from Pauli noise. In this paper, we present an efficient hard decoding algorithm for optimizing thresholds and lowering failure rates of an error-correcting code under general completely positive and trace-preserving (i.e., Markovian) noise. We use our hard decoding algorithm to study the performance of several errorcorrecting codes under various non-Pauli noise models by computing threshold values and failure rates for these codes. We compare the performance of our hard decoding algorithm to decoders optimized for depolarizing noise and show improvements in thresholds and reductions in failure rates by several orders of magnitude. Our hard decoding algorithm can also be adapted to take advantage of a code's non-Pauli transversal gates to further suppress noise. For example, we show that using the transversal gates of the 5-qubit code allows arbitrary rotations around certain axes to be perfectly corrected. Furthermore, we show that Pauli twirling can increase or decrease the threshold depending upon the code properties. Lastly, we show that even if the physical noise model differs slightly from the hypothesized noise model used to determine an optimized decoder, failure rates can still be reduced by applying our hard decoding algorithm.

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Section: E Effective Noise Channels For Concatenated Codesmentioning

confidence: 99%

Hard decoding algorithm for optimizing thresholds under general Markovian noise

Chamberland

Wallman

et al. 2017

Phys. Rev. A

Self Cite

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“…Since good logical magic states are hard to prepare in general, a magic-state distillation protocol [78] should be run beforehand to increase the quality of the noisy magic states. However, a large overhead cost comes with the distillation protocols and how to reduce the overhead has been under active research [79][80][81][82][83][84][85][86][87][88][89][90] (error correcting codes that avoid using the magic-state distillation have been also investigated [91][92][93][94][95][96][97][98]), and this costly nature of magic states motivates us to consider the resource theory of magic, which considers the "magicness" as precious resources.…”

Section: Fault-tolerant Quantum Computationmentioning

confidence: 99%

Universal limitations on implementing resourceful unitary evolutions

Takagi

Tajima

2020

Phys. Rev. A

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We derive a fundamental trade-off relation between accuracy of implementing a desired unitary evolution using a restricted set of free unitaries and the size of the assisting system, in terms of the resource generating/losing capacity of the target unitary. In particular, this relation implies that, for any theory that is equipped with a resource measure satisfying lenient conditions, any resource changing unitary cannot be perfectly implemented by a free unitary applied to system and environment if the environment has finite dimensions. Our results are applicable to a wide class of resources including energy, asymmetry, coherence, entanglement, and magic, imposing ultimate limitations inherent in those important physical settings, as well as providing new insights into operational restrictions in general resource theories.

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“…The desired values for δ depend on the particular fault-tolerant error correction protocol used to correct shift errors on encoded data qubits (see for instance [18][19][20][21][22]). For example, one can use a version of the error correction scheme (which reduces to Steane error correction for qubit CSS stabilizer codes) as proposed by Glancy and Knill in [9].…”

Section: Fault-tolerant Definitionsmentioning

confidence: 99%

“…In deriving equations (18)- (20), we kept only leading order terms in f  since the Hamiltonian in equation (14) neglects higher order terms in f and K. We keep only leading order terms since with current experimental parameter values, f  is roughly three orders of magnitude smaller than χ. Firstly, notice that the terms  ( ) R T introduce a relative rotation between the ñ |0 and ñ |1 state of the qubit. Neglecting the terms proportional to f  , it was shown in [3] that by choosing the total interaction time to be p c = T , the relative rotation between ñ |0 and ñ |1 can be set to one.…”

Section: Circuit Qed Implementationmentioning

confidence: 99%

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Fault-tolerant preparation of approximate GKP states

Shi

Chamberland²,

Cross³

2019

New J. Phys.

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Gottesman-Kitaev-Preskill (GKP) states appear to be amongst the leading candidates for correcting errors when encoding qubits into oscillators. However the preparation of GKP states remains a significant theoretical and experimental challenge. Until now, no clear definitions for fault-tolerantly preparing GKP states have been provided. Without careful consideration, a small number of faults can lead to large uncorrectable shift errors. After proposing a metric to compare approximate GKP states, we provide rigorous definitions of fault-tolerance and introduce a fault-tolerant phase estimation protocol for preparing such states. The fault-tolerant protocol uses one flag qubit and accepts only a subset of states in order to prevent measurement readout errors from causing large shift errors. We then show how the protocol can be implemented using circuit QED. In doing so, we derive analytic expressions which describe the leading order effects of the nonlinear dispersive shift and Kerr nonlinearity. Using these expressions, we show that to mitigate the nonlinear dispersive shift and Kerr terms would require the protocol to be implemented on time scales four orders of magnitude longer than the time scales relevant to the protocol for physically motivated parameters. Despite these restrictions, we numerically show that a subset of the accepted states of the fault-tolerant phase estimation protocol maintain good error correcting capabilities even in the presence of noise.

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Thresholds for Universal Concatenated Quantum Codes

Cited by 41 publications

References 22 publications

Hard decoding algorithm for optimizing thresholds under general Markovian noise

Hard decoding algorithm for optimizing thresholds under general Markovian noise

Universal limitations on implementing resourceful unitary evolutions

Fault-tolerant preparation of approximate GKP states

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