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arXiv:1311.2705 · IEEE Transactions on Information Theory

Quantum Stabilizer Codes From Maximal Curves

Lingfei Jin

Published Nov 19, 2013• 33 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1109/tit.2013.2287694

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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.
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.
Quantum error-correcting codes play the role of suppressing noise and decoherence in quantum systems by introducing redundancy. Some strategies can be used to improve the parameters of these codes. For example, entanglement can provide a way for quantum error-correcting codes to achieve higher rates than the one obtained by means of the traditional stabilizer formalism. Such codes are called entanglement-assisted quantum error-correcting (EAQEC) codes. In this paper, we utilize algebraic geometry codes to construct several families of EAQEC codes derived from the Euclidean and the Hermitian construction. Three families constructed here consist of codes whose quantum Singleton defect is equal to zero, one, or two. We also construct families of EAQEC codes with an encoding rate exceeding the quantum Gilbert-Varshamov bound. Additionally, asymptotically good towers of linear complementary dual codes are used to obtain asymptotically good families of EAQEC codes consuming maximal entanglement. Furthermore, a simple comparison with the quantum Gilbert-Varshamov bound demonstrates that, by utilizing the proposed construction, it is possible to generate an asymptotically family of EAQEC codes that exceeds this bound.
It is well known that n-length stabilizer quantum error correcting codes (QECCs) can be obtained via n-length classical error correction codes (CECCs) over GF(4), that are additive and self-orthogonal with respect to the trace Hermitian inner product. But, most of the CECCs have been studied with respect to the Euclidean inner product. In this paper, it is shown that n-length stabilizer QECCs can be constructed via 3n-length linear CECCs over GF(2) that are self-orthogonal with respect to the Euclidean inner product. This facilitates usage of the widely studied self-orthogonal CECCs to construct stabilizer QECCs. Moreover, classical, binary, self-orthogonal cyclic codes have been used to obtain stabilizer QECCs with guaranteed quantum error correcting capability. This is facilitated by the fact that (i) self-orthogonal, binary cyclic codes are easily identified using transform approach and (ii) for such codes lower bounds on the minimum Hamming distance are known. Several explicit codes are constructed including two pure MDS QECCs.
In the present paper, we show that if the dimension of an arbitrary algebraic geometry code over a finite field of even characteristic is slightly less than n/2-g with n being the length of the code and g being the genus of the base curve, then it is equivalent to an Euclidean self-orthogonal code. Previously, such results required a strong condition on the existence of a certain differential. We also show a similar result on Hermitian self-orthogonal algebraic geometry codes. As a consequence, we can apply our result to quantum codes and obtain some good quantum codes. In particular, we obtain a q-ary quantum [[q+1,1]]-MDS code for an even power q which is essential for quantum secret sharing.
In order to construct quantum error correction code, we consider additive codes over Galois field GF(4), which are self-orthogonal with respect to a certain Hermitian product. In this paper, we first propose a new approach to the construction of Hermitian self-orthogonal linear codes, applying the extension to get a longer length, and prove the codes have good minimum distance. Then, we investigate the corresponding quantum stabilizer codes from the classical codes. Six optimal quantum stabilizer codes have been achieved to show the effectiveness of the proposed construction.
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.