Quantum Error Correcting Codes Using Qudit Graph States

TL;DR

This paper extends graph state techniques to qudits of arbitrary dimension, constructing quantum error correcting codes with various distances, including some optimal and new codes, and generalizes stabilizer concepts.

Contribution

It introduces methods to construct qudit graph codes of various distances, including saturating bounds, and extends stabilizer formalism to general dimensions.

Findings

Codes of distance 2 saturate the quantum Singleton bound for large n and D.

Computer searches identified new codes with distances 3 and 4.

Stabilizer concepts are generalized to provide dual representations of additive codes.

Abstract

Graph states are generalized from qubits to collections of $n$ qudits of arbitrary dimension $D$, and simple graphical methods are used to construct both additive and nonadditive quantum error correcting codes. Codes of distance 2 saturating the quantum Singleton bound for arbitrarily large $n$ and $D$ are constructed using simple graphs, except when $n$ is odd and $D$ is even. Computer searches have produced a number of codes with distances 3 and 4, some previously known and some new. The concept of a stabilizer is extended to general $D$, and shown to provide a dual representation of an additive graph code.