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arXiv:1007.1778 · IEEE Transactions on Information Theory

Quantum Error Correction Beyond the Bounded Distance Decoding Limit

Kenta Kasai, Manabu Hagiwara, Hideki Imai, Kohichi Sakaniwa

Published Sep 21, 2011• 44 citations
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Abstract

In this paper, we consider quantum error correction over depolarizing channels with nonbinary low-density parity-check codes defined over Galois field of size 2p. The proposed quantum error correcting codes are based on the binary quasi-cyclic Calderbank, Shor, and Steane (CSS) codes. The resulting quantum codes outperform the best known quantum codes and surpass the performance limit of the bounded distance decoder. By increasing the size of the underlying Galois field, i.e., 2p, the error floors are considerably improved.

DOI
10.1109/tit.2011.2167593

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Quantum error correcting codes are essential to achieve fault-tolerant quantum computation. This work introduces two constructions of high-rate, dual-containing (DC) Calderbank--Shor--Steane low-density parity-check (LDPC) codes based on quasi-dyadic matrices. We characterize the automorphism group of such codes, investigate their minimum distance behavior, and provide several theoretical results on their cycle properties. Monte Carlo simulations under depolarizing and phenomenological noise show better finite-length logical error rates than the considered DC benchmark codes and competitive performance against several state-of-the-art quantum LDPC code families. Finally, we employ an automorphism-ensemble belief propagation decoder to improve their decoding performance.
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We construct a new family of Calderbank-Shor-Steane (CSS) codes using the generator and parity-check matrices of Low-Density Generator Matrix (LDGM) codes, with row operations applied to both matrices in order to achieve the desired quantum rate. Decoding is performed in an iterative manner, by applying message passing over the associated graph, and discrete Density Evolution (DDE) is used to optimize performance in the depolarizing channel. The proposed construction offers high flexibility and easiness in the design, producing quantum codes that possess excellent error correction capabilities. By properly designing the structure of the code, we are able to control and bound the weight of the stabilizer generators to a small value, which results in codes particularly well suited for fault-tolerant quantum computation. At the same time, these codes achieve very good performance in terms of error correction capability.
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Quantum Low Density Generator Matrix (QLDGM) codes based on Calderbank-Steane-Shor (CSS) constructions have shown unprecedented error correction capabilities in the paradigm of quantum communication. Recently, a strategy based on non-CSS quantum codes derived from QLDGM CSS codes has been shown to surpass other Quantum Low Density Parity Check (QLDPC) schemes proposed in the literature over the depolarizing channel. Given the importance of quantum channel estimation and the impact it has on the performance of QLDPC codes, in this article, we study the behaviour of non-CSS QLDGM codes under the umbrella of channel mismatch. We begin by showing how a relatively accurate estimate of the quantum channel is pivotal for these codes to perform appropriately. We follow this by analyzing an off-line and an on-line quantum channel parameter estimation technique, as well as discussing how these methods affect the Quantum Error Correction (QEC) codes under consideration. Finally, we show how the on-line methodology yields similar performance to the perfect channel knowledge scenario despite its relative simplicity.