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Fortschritte der Physik

The Quantum Hamming and Hexacodes

T. Beth, Markus Grassl

Published Jun 1, 1998• 21 citations
Publisher

Abstract

In this paper we present the relations between algebra and physics in the context of quantum error-correcting codes, showing the connection between Pauli matrices, finite fields, classical and quantum error-correcting codes. We show how to encode and especially decode (i.e., correct errors) quantum error-correcting codes. This is exemplarily demonstrated for the cyclic quantum Hamming code of length five and its extension, the quantum Hexacode.

DOI
10.1002/(sici)1521-3978(199806)46:4/5<459::aid-prop459>3.0.co;2-h

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In this paper we present the relations between algebra and physics in the context of quantum error-correcting codes, showing the connection between Pauli matrices, finite fields, classical and quantum error-correcting codes. We show how to encode and especially decode (i.e., correct errors) quantum error-correcting codes. This is exemplarily demonstrated for the cyclic quantum Hamming code of length five and its extension, the quantum Hexacode.
The goal of this paper is to provide a self contained detailed and rigorous mathematical introduction to some aspects of the quantum error-correcting codes and especially quantum stabilizer codes without venturing much, if at all, into the world of physics. While most of the results presented are not new, it is not easy to extract a precise mathematical formulation of results and to provide their rigorous proofs by reading the vast number of papers in the field, quite a few of which are written by computer scientists or physicists. It is this formulation and proofs - quite a few of of them new - that we present here. The connections between (a) quantum stabilizer codes and classical self-orthogonal codes and (b) between error correction and error detection have been established in details. Techniques from algebra of finite fields as well as representations of finite abelian groups have been employed here.
. We introduce a new class of quantum error--correcting codes derived from (classical) Reed--Solomon codes over finite fields of characteristic two. Quantum circuits for encoding and decoding based on the discrete cyclic Fourier transform over finite fields are presented. 1 Introduction During the last years it has been shown that computers taking advantage of quantum mechanical phenomena outperform currently used computers. The striking examples are integer factoring in polynomial time (see [18]) and finding pre--images of an n--ary Boolean function ("searching") in time O( p 2 n ) (see [12]). Quantum computers are not only of theoretical nature---there are several suggestions how to physically realize them (see, e. g., [6, 7]). On the way towards building a quantum computer, one very important problem is to stabilize quantum mechanical systems since they are very vulnerable. A theory of quantum error--correcting codes has already been established (see [15]). Nevertheless, the pr...
The goal of this paper is twofold. First, to provide a self contained, detailed and rigorous mathematical introduction to some aspects of the quantum error-correcting codes and especially quantum stabilizer codes and their connection to self-orthogonal linear codes. This has been done without venturing that much, if at all, into the world of physics. While most of the results presented are not new, it is not easy to extract a precise mathematical formulation of results and to provide their rigorous proofs by reading the vast number of papers in the field, quite a few of which are written by computer scientists or physicists. It is this formulation and proofs, some of which are new, that we present here. Techniques from algebra of finite fields as well as representations of finite abelian groups have been employed. The second goal is the construction of some stabilizer codes via self-orthogonal linear codes associated to algebraic curves.
I develop methods for analyzing quantum error-correcting codes, and use these methods to construct an infinite class of codes saturating the quantum Hamming bound. These codes encode k=n-j-2 quantum bits (qubits) in n=2j{2}^{\mathit{j}}2j qubits and correct t=1 error. \textcopyright{} 1996 The American Physical Society.