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Journal of Zhejiang University(Engineering Science)

Study on a quantum cyclic code for correcting two or more quantum errors

Xiao Chen

Published Jan 1, 2002• 0 citations

Abstract

The condition for constructing quantum cyclic codes of prime code length p with form8j+ 3was obtained via the quadratic residue method.A theorem on the form of quantum cyclic codes'generating ma- trix was proved.As an example,directexamination of digital computation showed thatthe quantum cyclic code for encoding 1 9qubits into 1 qubit and correcting 3errors satisfies error- correcting conditions.This code has never been reported in literature as far as we know.Another method for determining the already known quantum cyclic code for encoding1 1 qubits into1 qubit and correcting2 errors is given.Extended generating matrix method is proposed to express the error- correcting condition,which requires that the boolean summation of every four columns of the extended generating matrix should notbe an all zeros col- umn for codes correcting two errors.Meanwhile a quantum code should be self- orthogonal,that is the symplectic inner products of each pair of generating vectors should be zero.The codes we obtained satisfy all these conditions.

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The condition for constructing quantum cyclic codes of prime code length p with form8j+ 3was obtained via the quadratic residue method.A theorem on the form of quantum cyclic codes'generating ma- trix was proved.As an example,directexamination of digital computation showed thatthe quantum cyclic code for encoding 1 9qubits into 1 qubit and correcting 3errors satisfies error- correcting conditions.This code has never been reported in literature as far as we know.Another method for determining the already known quantum cyclic code for encoding1 1 qubits into1 qubit and correcting2 errors is given.Extended generating matrix method is proposed to express the error- correcting condition,which requires that the boolean summation of every four columns of the extended generating matrix should notbe an all zeros col- umn for codes correcting two errors.Meanwhile a quantum code should be self- orthogonal,that is the symplectic inner products of each pair of generating vectors should be zero.The codes we obtained satisfy all these conditions.

Quantum synchronizable codes are quantum error-correcting codes designed to correct the effects of both quantum noise and block synchronization errors. While it is known that quantum synchronizable codes can be constructed from cyclic codes that satisfy special properties, only a few classes of cyclic codes have been proved to give promising quantum synchronizable codes. In this paper, using quadratic residue codes and their supercodes, we give a simple construction for quantum synchronizable codes whose synchronization capabilities attain the upper bound. The method is applicable to cyclic codes of prime length.

The problem of finding stabilizer quantum-error-correcting codes can be transformed into the problem of finding classical self-orthogonal linear codes over the Galois field GF(4) under a Hermitian inner product.A classical cyclic code over the field GF(4) is self-orthogonal with respect to the Hermitian inner product iff the generator polynomial of its dual code is a factor of its generator polynomial.Based on this connection,a class of quantum cyclic codes is constructed by finding corresponding classical cyclic codes.Some examples are also given in detail.

We propose two types, namely Type-I and Type-II, quantum stabilizer codes using quadratic residue sets of prime modulus given by the form

p=4n\pm1

. The proposed Type-I stabilizer codes are of cyclic structure and code length

N=p

. They are constructed based on multi-weight circulant matrix generated from idempotent polynomial, which is obtained from a quadratic residue set. The proposed Type-II stabilizer codes are of quasi-cyclic (QC) structure and code length

N=pk

, where

k

is the size of a quadratic residue set. They are constructed based on structured sparse-graphs codes derived from proto-matrix and circulant permutation matrix. With the proposed methods, we design rich classes of cyclic and quasi-cyclic quantum stabilizer codes with variable code length. We show how the commutative constraint (also referred to as the Symplectic Inner Product constraint) for quantum codes can be satisfied for each proposed construction method. We also analyze both the dimension and distance for Type-I stabilizer codes and the dimension of Type-II stabilizer codes. For the cyclic quantum stabilizer codes, we show that they meet the existing distance bounds in literature.

In this paper, we consider a family of one-generator quasi-cyclic (QC) codes and their applications in quantum codes construction. We give a sufficient condition for one-generator QC codes to be self-orthogonal with respect to the symplectic inner product. As the computational results, five new binary quantum codes with parameters [[45,29,5]], [[63,36,7]], [[73,55,5]], [[105,71,7]], and [[105,72,7]] improving the best-known lower bounds on minimum distance in Grassl's code tables are constructed.

A new class of quantum stabilizer codes is presented.The problem of finding stabilizer quantum-error-correcting codes can be transformed into the problem of finding additive self-orthogonal codes over the Galois field GF(4) under a trace inner product.Based on this connection,one method for constructing quantum constacyclic codes is presented by finding self-orthogonal classical constacyclic codes over the field GF(4) under a trace inner product.Finally,some examples are given.As an application of this method,quantum Hamming codes are constructed.This class of quantum codes includes well-known quantum cyclic codes.