Nonbinary quantum codes

Abstract

We present several results on quantum codes over general alphabets (that is, in which the fundamental units may have more than two states). In particular, we consider codes derived from finite symplectic geometry assumed to have additional global symmetries. From this standpoint, the analogs of Calderbank-Shor-Steane codes and of GF(4)-linear codes turn out to be special cases of the same construction. This allows us to construct families of quantum codes from certain codes over number fields; in particular, we get analogs of quadratic residue codes, including a single-error-correcting code encoding one letter in five, for any alphabet size. We also consider the problem of fault-tolerant computation through such codes, generalizing ideas of Gottesman (see Phys. Rev. A, vol.57, no.1, p127-37, 1998).