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British Journal of Mathematics & Computer Science

On the Construction of Odd Length Quantum Codes

Manish K. Gupta, R. Narula, Divya Taneja

Published Jan 10, 2015• 2 citations
Publisher

Abstract

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.

DOI
10.9734/bjmcs/2015/15059

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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.
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.
It is reasonable to expect the theory of quantum codes to be simplified in the case of codes of minimum distance 2; thus it makes sense to examine such codes in the hopes that techniques that prove effective there will generalize. With this in mind, we present a number of results on codes of minimum distance 2. We first compute the linear programming bound on the dimension of such a code, then show that this bound can only be attained when the code either is of even length, or is of length 3 or 5. We next consider questions of uniqueness, showing that the optimal code of length 2 or 1 is unique (implying that the well-known one-qubit-in-five single-error correcting code is unique), and presenting nonadditive optimal codes of all greater even lengths. Finally, we compute the full automorphism group of the more important distance 2 codes, allowing us to determine the full automorphism group of any GF(4)-linear code.
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.
In this paper, we present a characterization of (binary and non-binary) quantum error-correcting codes. Based on this characterization, we introduce a method to construct p p -ary quantum codes using Boolean functions satisfying a system of certain quadratic relations. As a consequence of the construction, we are able to construct quantum codes of minimum distance 2 2 . In particular, we produce a class of binary quantum ( ( n , 2 n − 2 − 1 2 ( n − 1 ( n − 1 ) / 2 ) , 2 ) ) ((n,2^{n-2}-\frac 12{n-1\choose (n-1)/2},2)) -codes for odd length n ≥ 5 n\ge 5 . For n ≥ 11 n\ge 11 , this improves the result by Rains in Quantum codes of minimal distance two , 1999, showing the existence of binary quantum ( ( n , 3 ⋅ 2 n − 4 , 2 ) ) ((n,3\cdot 2^{n-4},2)) -codes for odd n ≥ 5 n\ge 5 . Moreover, our binary quantum
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.
Graph states are generalized from qubits to collections of nnn qudits of arbitrary dimension DDD, and simple graphical methods are used to construct both additive and nonadditive, as well as degenerate and nondegenerate, quantum-error-correcting codes. Codes of distance 2 saturating the quantum Singleton bound for arbitrarily large nnn and DDD are constructed using simple graphs, except when nnn is odd and DDD is even. Computer searches have produced a number of codes with distances 3 and 4, some previously known and some new. The concept of a stabilizer is extended to general DDD, and shown to provide a dual representation of an additive graph code.