Constructing Quantum Error-Correcting Codes for p-State Systems from Classical Error-Correcting Codes∗

Quantum error-correcting codes have attracted much attention. Among many research articles, the most general and systematic construction is the so called stabilizer code construction [6] or additive code construction [2], which constructs a quantum error-correcting code as an eigenspace of an Abelian subgroup S of the error group. Thereafter Calderbank et al. [3] proposed a construction of S from an additive code over the finite field F4 with 4 elements. These constructions work for tensor products of 2-state quantum systems. However Knill [8], [9] and Rains [13] observed that the construction [2], [6] can be generalized to n-state systems by an appropriate choice of the error basis. Rains [13] also generalized the construction [3] using additive codes over F4 to p-state quantum systems, but his generalization does not relate the problem of quantum code construction to classical error-correcting codes. We propose a construction of quantum error-correcting codes for p-state systems from classical error-correcting codes which is a generalization of [3]. Throughout this paper, p denotes a prime number andm a positive integer. This paper is organized as follows. In Sect. 2, we review the construction of quantum codes for nonbinary systems. In Sect. 3, we propose a construction of quantum codes for p-state systems from classical codes over Fp2 . In Sect. 4, we propose a construction of quantum codes for p-state systems from classical linear codes over Fp2m . In Sect. 5, we discuss a