Graphical nonbinary quantum error-correcting codes

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 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 ((6,2p²,3))₂ₚ for odd p, whose coding subspace is not of a dimension that is a power of the dimension of the physical subsystem.