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Journal of Physics A Mathematical and Theoretical

A stabilizer code for uncorrelated errors can correct spatially correlated ones

Tsuyoshi Hayashi, Toshihico Arimitsu, Sachiko Kitajima, Fumio Shibata

Published May 30, 2007• 3 citations
Publisher

Abstract

It is shown that errors on qubits caused by spatially correlated noise of quantum Brownian motion can be corrected with a stabilizer code (a quantum error correction code) and recovery operations prepared for uncorrelated noise without modifications. The analysis is made by means of the quantum stochastic Liouville equation approach which has been developed within the canonical operator formalism for dissipative systems called non-equilibrium thermo field dynamics. This approach yields a transparent procedure to derive completely positive maps describing errors on qubits under given statistics of noise.

DOI
10.1088/1751-8113/40/24/f04

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It is shown that errors on qubits caused by spatially correlated noise of quantum Brownian motion can be corrected with a stabilizer code (a quantum error correction code) and recovery operations prepared for uncorrelated noise without modifications. The analysis is made by means of the quantum stochastic Liouville equation approach which has been developed within the canonical operator formalism for dissipative systems called non-equilibrium thermo field dynamics. This approach yields a transparent procedure to derive completely positive maps describing errors on qubits under given statistics of noise.
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The framework of the canonical operator formalism for dissipative and/or stochastic quantum systems named non-equilibrium thermo field dynamics (NETFD) is presented in a self-contained manner. As an application of the formalism, a model of noisy-channel which is under the influence of spatially correlated quantum Brownian motion is investigated within the framework of NETFD. It is shown that errors due to spatially correlated noises can be corrected by the quantum error-correction code and error-correction procedure prepared for those for independent noises.
Coherent errors in stabilizer codes are often correlated across qubits and QEC cycles. Having a general analytical treatment of such noise would thus be extremely valuable. We derive here the exact logical channel induced by repeated QEC cycles under correlated coherent ZZZ noise, and develop a broadly general cumulant-expansion framework that yields a tractable expression for the noise-averaged logical infidelity. Crucially, this expression is non-perturbative in the noise, and applies to arbitrary stabilizer codes and correlation structures. It reveals a feature with no analogue in standard stochastic Pauli error models: the induced channel depends on which stabilizer eigenspace is chosen as the codespace. Exploiting this, we introduce protected stabilizer eigenspace (PROSE) encoding, an error-suppression strategy that selects the optimal codespace. We show that this eigenspace can be efficiently identified in many relevant situations. Further, when combined with logical Pauli twirling, PROSE matches or outperforms standard error suppression techniques (dynamical decoupling, Pauli twirling of physical qubits). We also show that noise correlations, usually assumed to be harmful to QEC, can instead be a resource: with the right encoding, even positive correlations reduce the logical infidelity below the uncorrelated baseline. Our results offer a new, broadly applicable lens on correlated coherent noise in stabilizer codes.
We consider quantum error correction against correlated noise using simple and concatenated Calderbank-Shor-Steane codes as well as nnn-qubit repetition codes. We characterize the performance of various codes by means of the fidelity following the error correction of a single logical qubit in a quantum register. For concatenated codes we find a threshold in the single-qubit error rate below which the encoded qubit is perfectly protected. The threshold depends on the correlation strength of the noise and goes to zero for perfect correlation. Finally, we concatenate the traditional error correcting codes with a decoherence free subspace and evaluate the performance over the whole range from uncorrelated noise to perfectly correlated noise.