A Combinatorial Approach to Quantum Error Correcting Codes

TL;DR

This paper introduces a combinatorial algorithm for determining the minimum sequence of operations to transform one graph coloring into another, which directly relates to assessing the performance of quantum error correcting codes.

Contribution

It presents an explicit algorithm for computing the distance of quantum error correcting codes via graph coloring transformations.

Findings

The algorithm efficiently computes the minimum operation sequence.

It provides a new method for evaluating quantum code performance.

Potential to aid in constructing improved quantum codes.

Abstract

Motivated from the theory of quantum error correcting codes, we investigate a combinatorial problem that involves a symmetric $n$-vertices colourable graph and a group of operations (colouring rules) on the graph: find the minimum sequence of operations that maps between two given graph colourings. We provide an explicit algorithm for computing the solution of our problem, which in turn is directly related to computing the distance (performance) of an underlying quantum error correcting code. Computing the distance of a quantum code is a highly non-trivial problem and our method may be of use in the construction of better codes.