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All the optimal stabilizer codes of distance 3

Sixia Yu, Ying Dong, Qing Chen

Published Jan 14, 2009• 3 citations
OpenAlex

Abstract

Abstract — Optimal quantum stabilizer codes of distance 3 are explicitly constructed for all lengths except for the following four families of lengths 8fm − {1, 2} and fm+2 − {2, 3} with fm = 4 m −1 3 and m ≥ 2 being integer, for which our codes are of the best parameters known and are only one logical qubit less than the quantum Hamming bound. The optimality of our codes is ensured by saturating either the quantum Hamming bound or a stronger bound for three families of lengths 8fm + {1, 2} and fm+2 − 1 with m ≥ 1 derived from the linear programming bound. For the lengths less than 128 three previously unknown codes [[36, 29, 3]], [[37, 30, 3]] and [[81, 73, 3]] have been found. Index Terms — quantum error correction, 1-error correcting stabilizer codes, quantum Hamming bound, linear programming bound, optimal codes I.

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Abstract — Optimal quantum stabilizer codes of distance 3 are explicitly constructed for all lengths except for the following four families of lengths 8fm − {1, 2} and fm+2 − {2, 3} with fm = 4 m −1 3 and m ≥ 2 being integer, for which our codes are of the best parameters known and are only one logical qubit less than the quantum Hamming bound. The optimality of our codes is ensured by saturating either the quantum Hamming bound or a stronger bound for three families of lengths 8fm + {1, 2} and fm+2 − 1 with m ≥ 1 derived from the linear programming bound. For the lengths less than 128 three previously unknown codes [[36, 29, 3]], [[37, 30, 3]] and [[81, 73, 3]] have been found. Index Terms — quantum error correction, 1-error correcting stabilizer codes, quantum Hamming bound, linear programming bound, optimal codes I.
The quantum Hamming bound was originally put forward as an upper bound on the parameters of nondegenerate quantum codes, but over the past few decades much work has been done to show that many degenerate quantum codes must also obey this bound. In this paper, we show that there is a Hamming-like bound stricter than the quantum Hamming bound that applies to degenerate ttt-error-correcting stabilizer codes of length greater than some positive integer N(t)N(t)N(t). We show that this bound holds for all single-error-correcting degenerate stabilizer codes, forcing all but a handful of optimal distance-3 stabilizer codes to be nondegenerate.
The most popular class of quantum error correcting codes is stabilizer codes. Binary quantum stabilizer codes have been well studied, and Calderbank, Rains, Shor and Sloane (July 1998) have constructed a table of upper bounds on the minimum distance of these codes using linear programming methods. However, not much is known in the case of nonbinary stabilizer codes. In this thesis, we establish a bridge between selforthogonal classical codes over the finite field containing q2 elements and quantum codes, extending and unifying previous work by Matsumoto and Uyematsu (2000), Ashikhmin and Knill (November 2001), Kim and Walker (2004). We construct a table of upper bounds on the minimum distance of the stabilizer codes using linear programming methods that are tighter than currently known bounds. Finally, we derive code construction techniques that will help us find new codes from existing ones. All these results help us to gain a better understanding of the theory of nonbinary stabilizer codes.
Linear programming approaches have been applied to derive upper bounds on the size of classical and quantum codes. In this paper, we derive similar results for general quantum codes with entanglement assistance by considering a type of split weight enumerator. After deriving the MacWilliams identities for these enumerators, we are able to prove algebraic linear programming bounds, such as the Singleton bound, the Hamming bound, and the first linear programming bound. Our Singleton bound and Hamming bound are more general than the previous bounds for entanglement-assisted quantum stabilizer codes. In addition, we show that the first linear programming bound improves the Hamming bound when the relative distance is sufficiently large. On the other hand, we obtain additional constraints on the size of Pauli subgroups for quantum codes, which allow us to improve the linear programming bounds on the minimum distance of quantum codes of small length. In particular, we show that there is no [[27, 15, 5]] or [[28, 14, 6]] stabilizer code. We also discuss the existence of some entanglement-assisted quantum stabilizer codes with maximal entanglement. As a result, the upper and lower bounds on the minimum distance of maximal-entanglement quantum stabilizer codes with length up to 20 are significantly improved.
In this paper, we endeavor for an extensive study of [[n,n-3,2]] codes of odd length. We begin with the computation of the linear programming bound on the dimension of distance 2 codes of odd length and show that the [[n,n-3,2]] codes are optimal. We next find their generator matrix, stabilizer structure and also show that these codes are impure or degenerate except the [[3,0,2]] code which is pure by convention. In degenerate codes, distinct errors do not necessarily take the code space to orthogonal space. So sometimes they can correct more errors than that they can identify and has the capacity to store more information than a nondegenerate code. The present paper also establishes the existence of ((2m+1,2^(2m-2),2)) codes from the ((2m,2^(2m-2),2)) codes for all m>1. We have also constructed another class of distance 2 codes which are constructed using distance 3 codes.
Stabilizer codes form a special class of quantum error correcting codes. Nonbinary quantum stabilizer codes are studied in this thesis. A lot of work on binary quantum stabilizer codes has been done. Nonbinary stabilizer codes have received much less attention. Various results on binary stabilizer codes such as various code families and general code constructions are generalized to the nonbinary case in this thesis. The lower bound on the minimum distance of a code is nothing but the minimum distance of the currently best known code. The focus of this research is to improve the lower bounds on this minimum distance. To achieve this goal, various existing quantum codes are studied that have good minimum distance. Some new families of nonbinary stabilizer codes such as quantum BCH codes are constructed. Different ways of constructing new codes from the existing ones are also found. All these constructions together help improve the lower bounds.
One formidable difficulty in quantum communication and computation is to protect information-carrying quantum states against undesired interactions with the environment. To address this difficulty, many good quantum error-correcting codes have been derived as binary stabilizer codes. Fault-tolerant quantum computation prompted the study of nonbinary quantum codes, but the theory of such codes is not as advanced as that of binary quantum codes. This paper describes the basic theory of stabilizer codes over finite fields. The relation between stabilizer codes and general quantum codes is clarified by introducing a Galois theory for these objects. A characterization of nonbinary stabilizer codes overbfFqbf F_qbfFq​in terms of classical codes over$ bf F_q^2isprovidedthatgeneralizesthewell−knownnotionofadditivecodesoveris provided that generalizes the well-known notion of additive codes overisprovidedthatgeneralizesthewell−knownnotionofadditivecodesover bf F_4$of the binary case. This paper also derives lower and upper bounds on the minimum distance of stabilizer codes, gives several code constructions, and derives numerous families of stabilizer codes, including quantum Hamming codes, quadratic residue codes, quantum Melas codes, quantum Bose–Chaudhuri–Hocquenghem (BCH) codes, and quantum character codes. The puncturing theory by Rains is generalized to additive codes that are not necessarily pure. Bounds on the maximal length of maximum distance separable stabilizer codes are given. A discussion of open problems concludes this paper.