Symplectic Inner Product Research Articles

New construction of binary and nonbinary quantum stabilizer codes based on symmetric matrices

In this paper, we propose two construction methods for binary and nonbinary quantum stabilizer codes based on symmetric matrices. In the first construction, we use the identity and symmetric matrices to generate parity-check matrices that satisfy the symplectic inner product (SIP) for the construction of quantum stabilizer codes. In the second construction, we modify the first construction to generate parity-check matrices based on the Calderbank–Shor–Stean structure for the construction of quantum stabilizer codes. The binary and nonbinary quantum stabilizer codes whose parameters achieve equality of the quantum singleton bound are investigated with the code lengths ranging from 4 to 12.

Design of Quantum LDPC Codes From Quadratic Residue Sets

We design classes of quantum low-density parity-check (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 $p = 4n\pm 1$ . 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.

Efficient Quantum Stabilizer Codes: LDPC and LDPC-Convolutional Constructions

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 <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">)</i> ), and some of them perform as well as or better than the existing quantum codes.