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IEEE Transactions on Communications

Design of Quantum LDPC Codes From Quadratic Residue Sets

Yixuan Xie, Jinhong Yuan, Qifu Tyler Sun

Published Apr 17, 2018• 14 citations
Publisher

Abstract

We design classes of quantum low-density paritycheck (LDPC) codes, called quasi-cyclic stabilizer (QCS) codes, from conventional QC-LDPC codes. The proposed QCS codes belong to the family of non-Calderbank-Shor-Steane stabilizer codes. The QC-LDPC codes are self-orthogonal with respect to the symplectic inner product (SIP) and are constructed from submatrices of nonorthogonal Latin squares via array dispersion. The Latin squares are constructed using quadratic (non)-residue sets of prime modulus p, where p = 4n ± 1. For p = 4n - 1, two constructions, namely, Type-I-A and Type-I-B QCS codes, are proposed based on matrix superposition and matrix concatenation, respectively. For p = 4n +1, Type-II QCS codes are proposed based on permutations of a base matrix. We show that the parity-check matrix for Type-I-B and Type-II QCS codes is self-orthogonal with respect to the SIP for all orders of circulant permutation matrix. This resulting in ensembles of QCS codes characterized by a single base matrix. We show that the minimum distance of Type-II QCS codes can be lower bounded by the minimum distance of the QC-LDPC codes. Simulation results show that the proposed QCS codes outperform some codes in the literature with a noteworthy low-error floor, below 10-7, over quantum depolarizing channels.

DOI
10.1109/tcomm.2018.2827945

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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.
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We propose two constructions, namely Construction 1 and Construction 2, for the design of protograph quantum LDPC codes by exploiting conventional quasi cyclic protograph LDPC codes with lift operation, and tensor product operation, respectively. We show that by carefully designing the proto-matrix using quadratic residue sets of parameters p = 4n ± 1, the proposed Construction 1 yields protograph quantum LDPC codes of various rates. Furthermore, by performing tensor product between two non-binary parity-check matrices obtained from idempotent polynomials of a quadratic residue set, the proposed Construction 2 yields a class of protograph quantum LDPC codes with rate at least 0.9. The performance of the proposed codes over quantum depolarizing channel with iterative sum-product decoding algorithm is shown.