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Physical Review A

Approach for the construction of non-Calderbank-Steane-Shor low-density-generator-matrix–based quantum codes

Patricio Fuentes, Josu Etxezarreta Martínez, Pedro M. Crespo, Javier Garcia‐Frias

Published Jul 22, 2020• 15 citations
Publisher

Abstract

Quantum low-density generator matrix (QLDGM) codes based on Calderbank-Steane-Shor (CSS) constructions have shown unprecedented error correction capabilities, displaying much improved performance in comparison to other sparse-graph codes. However, the nature of CSS designs and the manner in which they must be decoded limit the performance that is attainable with codes that are based on this construction. This motivates the search for quantum code design strategies capable of avoiding the drawbacks associated with CSS codes. In this article, we introduce non-CSS quantum code constructions based on classical LDGM codes. The proposed codes are derived from CSS QLDGM designs by performing specific row operations on their quantum parity check matrices to modify the associated decoding graphs. The application of this method results in performance improvements in comparison to CSS QLDGM codes, while also allowing for greater flexibility in the design process. The proposed non-CSS QLDGM scheme outperforms the best quantum low-density parity check codes that have appeared in the literature.

DOI
10.1103/physreva.102.012423

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Quantum low-density generator matrix (QLDGM) codes based on Calderbank-Steane-Shor (CSS) constructions have shown unprecedented error correction capabilities, displaying much improved performance in comparison to other sparse-graph codes. However, the nature of CSS designs and the manner in which they must be decoded limit the performance that is attainable with codes that are based on this construction. This motivates the search for quantum code design strategies capable of avoiding the drawbacks associated with CSS codes. In this article, we introduce non-CSS quantum code constructions based on classical LDGM codes. The proposed codes are derived from CSS QLDGM designs by performing specific row operations on their quantum parity check matrices to modify the associated decoding graphs. The application of this method results in performance improvements in comparison to CSS QLDGM codes, while also allowing for greater flexibility in the design process. The proposed non-CSS QLDGM scheme outperforms the best quantum low-density parity check codes that have appeared in the literature.
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.
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.
Quantum low-density parity-check (QLDPC) codes provide a practical balance between error-correction capability and implementation complexity in quantum error correction (QEC). In this paper, we propose an algebraic construction based on dyadic matrices for designing both classical and quantum LDPC codes. The method first generates classical binary quasi-dyadic LDPC codes whose Tanner graphs have girth 6. It is then extended to the Calderbank-Shor-Steane (CSS) framework, where the two component parity-check matrices are built to satisfy the compatibility condition required by the recently introduced CAMEL-ensemble quaternary belief propagation decoder. This compatibility condition ensures that all unavoidable cycles of length 4 are assembled in a single variable node, allowing the mitigation of their detrimental effects by decimating that variable node.
Stabilizer codes are often constructed within the Calderbank--Shor--Steane (CSS) framework, where two mutually orthogonal binary classical codes define XXX and ZZZ-type stabilizer generators. While this structure is algebraically convenient, additional non-CSS constraints may help suppress low-weight logical operators and improve decoding performance in the finite-length regime. We thus introduce quantum XYZ stabilizer codes, whose parity-check matrix (PCM) is built from three pairwise orthogonal binary PCMs associated with XXX-, YYY-, and ZZZ-type stabilizer generators. A nontrivial point is that an XYZ code instance is not automatically genuinely non-CSS: the same stabilizer group may admit a CSS generating set. We characterize this collapse, obtaining algebraic and rank conditions for deciding when the YYY-type checks are redundant and when they define genuinely non-CSS stabilizer constraints. We also derive upper and lower bounds on the quantum minimum distance, including bounds for mixed Pauli logical operators. The novel framework includes a known non-CSS topological code, namely the XYZ2^22 hexagonal code, and yields also sparse finite-length quantum low-density parity-check (qLDPC) constructions from intersecting-subset and quasi-dyadic code families. Simulations under depolarizing code-capacity noise and quaternary belief propagation decoding show that the proposed XYZ qLDPC instances can outperform representative CSS qLDPC instances with similar finite-length parameters.
Quantum low-density-generator-matrix (QLDGM) codes are known to exhibit great error correction capabilities, surpassing existing quantum low-density-parity-check (QLDPC) codes and other sparse-graph schemes over the depolarizing channel. Most of the research on QLDPC codes and quantum error correction (QEC) is conducted for the symmetric instance of the generic Pauli channel, which incurs bit flips, phase flips, or a combination of both with the same probability. However, due to the behavior of the materials they are built from, some quantum devices must be modelled using a different channel model capable of accurately representing asymmetric scenarios in which the likelihood of a phase flip is higher than that of a bit flip. In this work, we study the design of QLDGM CSS codes for such Pauli channels. We show how codes tailored to the depolarizing channel are not well suited to these asymmetric environments and we derive methods to aptly design QLDGM CSS codes for this paradigm.
In this work, we propose a novel design of high-rate Calderbank--Shor--Steane (CSS) quantum low-density parity-check (QLDPC) codes based on quasi-dyadic matrices. The Tanner graphs associated with the two component classical codes used to construct the proposed CSS codes are characterized by a girth of at least 666. We also derive an exact count of length-444 cycles in the quaternary representation of the parity-check matrix. Monte Carlo simulations under code-capacity and phenomenological noise models show that the proposed codes have competitive finite-length logical error rate performance compared with state-of-the-art codes in both these settings.