Nonbinary Stabilizer Codes Over Finite Fields

Abstract

One formidable difficulty in quantum communication and computation is to protect information-carrying quantum states against undesired interactions with the environment. In past years, many good quantum error-correcting codes had 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 over Fq in terms of classical codes over F q 2 is provided that generalizes the well-known notion of additive codes over F4 of the binary case. This paper 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 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.

Quantum-error-correcting codes using qudit graph states

“…The ((11, 3 6 = 729, 3)) 3 code we found, the best possible additive code according to the linear programming bound in [24], falls short by a factor of 3 of saturating the K = 3 7 = 2187 QS bound, and even a nonadditive code based on this graph must have K ≤ 1990 [28].…”

“…It has been extended to cases where D is prime or a prime power in [6,24,30]. In [8] stabilizer codes were extended in a very general fashion to arbitrary D from a point of view that includes encoding.…”

“…• d -Indicates this is not a QS code, but the largest possible additive (graph or other) code for the given n, δ and D, This follows from linear programming bounds in [23] for D = 2 and [24] for D = 3, along with the fact, Sec. III B, that for an additive code, K must be an integer power of D when D is prime.…”

Some Hermitian Dual Containing BCH Codes and New Quantum Codes

“…It is well known that there is a close relationship between cyclotomic cosets and cyclic codes, see [9,11]. This suggests us to use q 2 -cyclotomic cosets of modulo n to characterize BCH codes over F q 2 , see [14].…”

“…The following Theorem 2.4 (given in [3,11] is well-known for constructing q-ary quantum codes from Hermitian dual containing (or self-orthogonal) codes over…”