Graphical nonbinary quantum error-correcting codes

Abstract

In this paper, we present a systematic way based on the nonbinary graph state of constructing good nonbinary quantum codes, both additive and nonadditive, for systems with integer dimensions. With a computer search, which results in many interesting codes including some nonadditive codes meeting the Singleton bounds, we are able to construct explicitly four families of optimal codes, namely, ${[[6,2,3]]}_{p}$, ${[[7,3,3]]}_{p}$, ${[[8,2,4]]}_{p}$, and ${[[8,4,3]]}_{p}$ for any odd dimension $\mathit{p}$ and a family of nonadditive codes ${((5,p,3))}_{p}$ for arbitrary $p>3$. In the case of composite numbers as dimensions, we also construct a family of stabilizer codes ${((6,2{p}^{2},3))}_{2p}$ for odd $\mathit{p}$, whose coding subspace is not of a dimension that is a power of the dimension of the physical subsystem.