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International Journal of Modern Physics B

A novel construction for quantum stabilizer codes based on binary formalism

Duc Manh Nguyen, Sunghwan Kim

Published Mar 24, 2020• 16 citations
Publisher

Abstract

In this research, we propose a novel construction of quantum stabilizer code based on a binary formalism. First, from any binary vector of even length, we generate the parity-check matrix of the quantum code from a set composed of elements from this vector and its relations by shifts via subtraction and addition. We prove that the proposed matrices satisfy the condition constraint for the construction of quantum codes. Finally, we consider some constraint vectors which give us quantum stabilizer codes with various dimensions and a large minimum distance with code length from six to twelve digits.

DOI
10.1142/s0217979220500599

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In this research, we propose a novel construction of quantum stabilizer code based on a binary formalism. First, from any binary vector of even length, we generate the parity-check matrix of the quantum code from a set composed of elements from this vector and its relations by shifts via subtraction and addition. We prove that the proposed matrices satisfy the condition constraint for the construction of quantum codes. Finally, we consider some constraint vectors which give us quantum stabilizer codes with various dimensions and a large minimum distance with code length from six to twelve digits.
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.
In this paper, a new but simple construction of stabilizer codes and related entanglement-assisted quantum error-correcting codes is proposed based on syndrome assignment by classical parity-check matrices. This method turns the construction of quantum stabilizer codes to the construction of classical parity-check matrices satisfying a specific commutative condition. The designed minimum distance 2t*+1 of the constructed quantum stabilizer codes can be achieved by a commutative classical parity-check matrix with classical minimum distance 4t*-m, where the parameterm, 0 ≤m≤ 2t*, depends on a property of the parity-check matrix. Asmdecreases, there is an increasing set of additional correctable error operators beyond the designed error correcting capability t*. The (asymptotic) coding efficiency is at least comparable to that of CSS codes. A class of quantum Reed-Muller codes is constructed and codes in this class have a larger set of correctable error operators than that of the quantum Reed-Muller codes previously developed in the literature. Quantum circulant codes are also constructed and many of them are optimal in terms of their coding parameters.
In this paper, new conditions on parameters in difference sets are derived to satisfy symplectic inner product, and new constructions of quantum stabilizer codes are proposed from the conditions. The conversion of the difference sets into parity-check matrices is first explained. Then, the proposed code construction is composed of three steps, which are to choose the generators of quantum stabilizer code, to determine the quantum stabilizer groups, and to determine subspace codewords with large minimum distance. The quantum stabilizer codes with various length are also presented to explain the practicality of the code construction. The proposed design can be applied to quantum stabilizer code construction based on combinatorial design.
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.
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.
In this paper, we propose quantum stabilizer codes based on a new construction of self-orthogonal trace-inner product codes over the Galois field with 4 elements (GF(4)). First, from any two binary vectors, we construct a generator matrix of linear codes whose components are over GF(4). We prove that the proposed linear codes comply with the self-orthogonal, trace-inner product. Then, we propose mapping tables to construct new quantum stabilizer codes by using linear codes. Comparison results show that our proposed quantum codes have various dimensions for any code length with the capacity for better errors correction relative to the referenced quantum codes.