On the relation between a graph code and a graph state

TL;DR

This paper explores the relationship between graph states and graph codes derived from the same graph, revealing that a graph state can be expressed as a superposition of logical qubits of the related graph code, and discusses applications in stabilizer code analysis.

Contribution

It establishes a direct relation between graph states and graph codes, providing new methods for analyzing stabilizer codes and their local equivalences.

Findings

A graph state is a superposition of logical qubits of the related graph code.

Local complementation can be used to find locally equivalent stabilizer codes.

A method to determine the stabilizer group of a graph code is proposed.

Abstract

A graph state and a graph code respectively are defined based on a mathematical simple graph. In this work, we examine a relation between a graph state and a graph code both obtained from the same graph, and show that a graph state is a superposition of logical qubits of the related graph code. By using the relation, we first discuss that a local complementation which has been used for a graph state can be useful for searching locally equivalent stabilizer codes, and second provide a method to find a stabilizer group of a graph code.