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International Conference on Ubiquitous and Future Networks

Construction and complement circuit of a quantum stabilizer code with length 7

Duc Manh Nguyen, Sunghwan Kim

Published Jul 1, 2016• 9 citations
Publisher

Abstract

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.

DOI
10.1109/icufn.2016.7537043

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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 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.
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.