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NATO science for peace and security series. D, Information and communication security

Error Correcting Quantum Codes and Algebraic Curves

Artur Elezi

Published Jan 1, 2015• 0 citations
Publisher

Abstract

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.

DOI
10.3233/978-1-61499-520-3-286

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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.
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.
There has been a lot of effort to construct good quantum codes from the classical error correcting codes. Constructing new quantum codes, using Hermitian self-orthogonal codes, seems to be a difficult problem in general. In this paper, Hermitian self-orthogonal codes are studied from algebraic function fields. Sufficient conditions for the Hermitian self-orthogonality of an algebraic geometry code are presented. New Hermitian self-orthogonal codes are constructed from projective lines, elliptic curves, hyper-elliptic curves, Hermitian curves, and Artin-Schreier curves. In addition, over the projective lines, we construct new families of MDS quantum codes with parameters [[N,N−2K,K+1]]q[[N,N-2K,K+1]]_q[[N,N−2K,K+1]]q​ under the following conditions: i) N=t(q−1)+1N=t(q-1)+1N=t(q−1)+1 or t(q−1)+2t(q-1)+2t(q−1)+2 with t∣(q+1)t|(q+1)t∣(q+1) and K=⌊t(q−1)+12t⌋+1K=\lfloor\frac{t(q-1)+1}{2t}\rfloor+1K=⌊2tt(q−1)+1​⌋+1; ii) (n−1)∣(q2−1)(n-1)|(q^2-1)(n−1)∣(q2−1), N=nN=nN=n or N=n+1N=n+1N=n+1, K0=⌊n+q−1q+1⌋K_0=\lfloor\frac{n+q-1}{q+1}\rfloorK0​=⌊q+1n+q−1​⌋, and K≥K0+1K\ge K_0+1K≥K0​+1; iii) N=tq+1N=tq+1N=tq+1, ∀ 1≤t≤q\forall~1\le t\le q∀ 1≤t≤q and K=⌊tq+q−1q+1⌋+1K=\lfloor\frac{tq+q-1}{q+1}\rfloor+1K=⌊q+1tq+q−1​⌋+1; iv) n∣(q2−1)n|(q^2-1)n∣(q2−1), n2=ngcd⁡(n,q+1)n_2=\frac{n}{\gcd (n,q+1)}n2​=gcd(n,q+1)n​, ∀ 1≤t≤q−1n2−1\forall~ 1\le t\le \frac{q-1}{n_2}-1∀ 1≤t≤n2​q−1​−1, N=(t+1)n+2N=(t+1)n+2N=(t+1)n+2 and K=⌊(t+1)n+1+q−1q+1⌋+1K=\lfloor \frac{(t+1)n+1+q-1}{q+1}\rfloor+1K=⌊q+1(t+1)n+1+q−1​⌋+1.
Research in the field of quantum algorithms and quantum error correction is progressing at an astounding rate. There are many good papers on both subjects, but reading even a few of these may seem a daunting task to the newcomer. The aim of this paper is to give a leisurely introduction to the basic theory of quantum error correcting codes without appealing to even the most basic notions in physics. Thus the article is not a substitute for important papers such as [12] or [7] but rather an advertisement for them. I would be pleased if, in addition, some readers view this as a useful companion article if and when they go on to read more substantial literature on the subject of quantum error correction. I present nothing new here. Rather, I give an elementary account of the important theorems and proofs which appear in these fundamental works using only undergraduate algebra and a bit of classical coding theory. In particular, I give a full proof of the Knill/Laflamme theorem as well as an elementary treatment of stabilizer codes. The goal is to make the literature dealing with this exciting new area more accessible to discrete mathematicians.
Matrix-product codes over finite fields are an important class of long linear codes by combining several commensurate shorter linear codes with a defining matrix over finite fields. The construction of matrix-product codes with certain self-orthogonality over finite fields is an effective way to obtain good qqq-ary quantum codes of large length. This article has two purposes: the first is to summarize some results of this topic obtained by the author of this article and his cooperators in [10-12]; the second is to add some new results on quasi-orthogonal matrices (resp. quasi-unitary matrices), Euclidean dual-containing (resp. Hermitian dual-containing) matrix-product codes and qqq-ary quantum codes derived from these matrix-product codes.
Quantum computing’s seemingly perpetual promise of nearness still has a few hurdles \nto surmount before it can become a reality. Among these hurdles is that of protection of \ninformation from random errors. A potential solution for this challenge is stabilizer codes, \nwhich are the analog of classical linear error-correcting codes, however, with an additional \naxis of error possibility. By and large, quantum computing is discussed in the standard \nqubit, or two-level, language, however, it is worth considering the case of qudits, or more \nthan two-level systems. Often times results follow simply from some form of algebraic \nextension: typically group theory or linear algebra. In this work we consider some features \nthat are not immediately apparent from that approach, more often appealing to physical \nintuition to guide our mathematical ideas, then proving these ideas using the language of \nqudit operators. \n \nHere we consider two particular previously unexplored ideas. The first idea is that of \nembedding and inscribing of codes into spaces of different sizes than the stabilizer code \nwas originally designed for. Here, we show that all codes can be embedded, and that for \ninfinitely many primes we can in fact guarantee that the distance is at least preserved–a \nsomewhat surprising result. The second idea is, in a way an application of those presented \nin the first idea, that of turning many stabilizer codes into hybrid codes by taking advantage \nof relabelling of syndromes and the rapid increase in syndrome space upon using codes in \nlarge spaces. Both of these are somewhat useful in their current forms, however, with some \nadditional mathematical work could be turned into potentially very powerful tools for the \nprotection of quantum information. We finish off by, and along the way, mentioning various \nfuture directions to carry work on this.
A curve attaining the Hasse-Weil bound is called a maximal curve. Usually, classical error-correcting codes obtained from a maximal curve have good parameters. However, the quantum stabilizer codes obtained from such classical error-correcting codes via Euclidean or Hermitian self-orthogonality do not always possess good parameters. In this paper, the Hermitian self-orthogonality of algebraic geometry codes obtained from two maximal curves is investigated. It turns out that the stabilizer quantum codes produced from such Hermitian self-orthogonal classical codes have good parameters.