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IEEE Transactions on Communications

Construction of Multiple-Rate Quantum LDPC Codes Sharing One Scalable Stabilizer Circuit

Yun-Jiang Wang, Zhuo-Yan Xiao, Yi Zhang, Xing-Yu Xiong, Sha Shi

Published Dec 23, 2022• 2 citations
Publisher

Abstract

Variable-rate coding schemes that support a variety of different rates while maintaining the same fundamental encoder/decoder architectures are of great interest in practical communication systems. Similar error-correcting code schemes are also in demand in quantum settings. However, generally, it is difficult to introduce the variable-rate coding schemes into the quantum coding domain attributed to two challenges: Obtaining new quantum codes from old ones systematically and sharing encoder-decoder components among the newly obtained quantum codes. In this correspondence, a multiple-rate coding scheme is introduced into the quantum coding domain, we achieve this goal by providing a systematic method to construct new non-homogeneous quantum LDPC codes of CSS type from an old one (named as the mother code) whose classical parity-check matrix (PCM) is row-circulant. The basic idea is to split rows of the highest-rate stabilizer elaborately to produce the stabilizer for lower rates, which ensures that the generated code family owns the same code length. More importantly, thanks to the nested stabilizer structures possessed by the resulting quantum codes, our method also enables a scalable stabilizer circuit to be shared among them.

DOI
10.1109/tcomm.2022.3231879

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Variable-rate coding schemes that support a variety of different rates while maintaining the same fundamental encoder/decoder architectures are of great interest in practical communication systems. Similar error-correcting code schemes are also in demand in quantum settings. However, generally, it is difficult to introduce the variable-rate coding schemes into the quantum coding domain attributed to two challenges: Obtaining new quantum codes from old ones systematically and sharing encoder-decoder components among the newly obtained quantum codes. In this correspondence, a multiple-rate coding scheme is introduced into the quantum coding domain, we achieve this goal by providing a systematic method to construct new non-homogeneous quantum LDPC codes of CSS type from an old one (named as the mother code) whose classical parity-check matrix (PCM) is row-circulant. The basic idea is to split rows of the highest-rate stabilizer elaborately to produce the stabilizer for lower rates, which ensures that the generated code family owns the same code length. More importantly, thanks to the nested stabilizer structures possessed by the resulting quantum codes, our method also enables a scalable stabilizer circuit to be shared among them.
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.
This paper proposes efficient construction of new classes of quantum low-density parity-check (LDPC) codes through quasi-cyclic matrices, symmetric matrices and scrambling matrices. Unlike many of the existing quantum LDPC codes, the proposed construction is based on classic binary codes rather than quaternary codes, targets the general stabilizer formalism rather than the special CSS formalism, employs systematic approaches rather than computer search, and generates a rich class of codes of different rates and lengths rather than a single code. Simulations demonstrate that these new codes outperform the existing non-CSS LDPC stabilizer codes reported in literature.
Classical row-circulant quasi-cyclic (QC) low-density parity check (LDPC) matrices are known to generate efficient high-rate short and moderate-length QC-LDPC codes, while the comparable random structures exhibit numerous short cycles of length-4. Therefore, we conceive a general formalism for constructing nondual-containing Calderbank-Shor-Steane (CSS)-type quantum low-density parity check (QLDPC) codes from arbitrary classical row-circulant QC-LDPC matrices. We apply our proposed formalism to the classical balanced incomplete block design (BIBD)-based row-circulant QC-LDPC codes for demonstrating that our designed codes outperform their dual-containing CSS-type counterparts as well as the entanglement-assisted (EA)-QLDPC codes.
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.
In this paper, a new method for the construction of a quantum stabilizer code from circulant permutation matrices is discussed. First, we choose a finite-length vector randomly, and we can construct circulant permutation matrices from the vectors. Then, the parity-check matrix can be produce from the circulant permutation matrices. Hence, the generators of stabilizer code are determined according to the parity-check matrix and quantum stabilizer group are defined from the generators. From the stabilizer group, codewords of the proposed quantum codes can also be generated. Finally, a complete efficient encoding and decoding quantum circuit of [[7,1,3]] is proposed. [[7,1,3]] is stabilizer code that construction based on our method is an seven-qubit code that protects a one-qubit state with up to one error, which is very important for quantum information processing.
Existing quantum stabilizer codes constructed from the classic binary codes exclusively belong to the special subclass of Calderbank-Shor-Steane (CSS) codes. This paper fills in the gap by proposing five systematic constructions for non-CSS stabilizer codes, the first four of which are based on classic binary quasi-cyclic low-density parity-check (LDPC) codes and last on classic binary LDPC-convolutional codes. These new constructions exploit structured sparse graphs, make essential use of simple and powerful coding techniques including concatenation, rotation and scrambling, and generate rich classes of codes with a wide range of lengths and rates. We derive the sufficient, and in some cases also the necessary, conditions for each construction to satisfy the general symplectic inner product (SIP) condition, and develop practical decoder algorithms for these codes. The resulting codes are the first classes of non-CSS quantum LDPC codes and non-CSS quantum convolutional codes rooted from classic binary codes (rather than codes in GF(4)), and some of them perform as well as or better than the existing quantum codes.