Graphical Nonbinary Quantum Error-Correcting Codes

TL;DR

This paper introduces a systematic method for constructing nonbinary quantum error-correcting codes using graph states, resulting in several optimal and nonadditive codes for systems with various dimensions.

Contribution

It presents a new systematic construction approach for nonbinary quantum codes based on graph states, including explicit families of optimal and nonadditive codes for different dimensions.

Findings

Constructed four families of optimal codes meeting Singleton bounds.

Developed a family of nonadditive codes for arbitrary p>3.

Built stabilizer codes with non-power-of-dimension coding subspaces.

Abstract

In this paper, based on the nonbinary graph state, we present a systematic way of constructing good non-binary quantum codes, both additive and nonadditive, for systems with integer dimensions. With the help of 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 code $((5,p,3))_p$ for arbitrary $p>3$. In the case of composite numbers as dimensions, we also construct a family of stabilizer codes $((6,2\cdot p^2,3))_{2p}$ for odd $p$, whose coding subspace is {\em not} of a dimension that is a power of the dimension of the physical subsystem.