Withdrawn: A Measurement-based Algorithm for Graph Colouring

TL;DR

This paper proposes a quantum algorithm for graph coloring that uses quantum measurements and resets, showing polynomial scaling in edges but exponential in vertices, though the work has been withdrawn due to runtime misinterpretation.

Contribution

It introduces a quantum approach to graph coloring with a novel measurement and reset technique, and provides numerical evidence of its scaling properties.

Findings

Runtime scales exponentially with vertices.

Average runtime scales polynomially with edges.

The work has been withdrawn due to a misinterpretation of runtime complexity.

Abstract

In a previous version of this document we misinterpreted the runtime of a part of the described algorithm. Indeed, the runtime is not better than the Grover-Algorithm. We therefor withdraw this work. We present a novel algorithmic approach to find a proper vertex colouring of a graph with $d$ colours, if it exists. We associate a $d$-dimensional quantum system with each vertex and the initial state is a mixture of all possible colourings, from which we obtain a random proper colouring of the graph by measurements. The non-deterministic nature of the quantum measurement is tackled by a reset operation, which can revert the effect of unwanted projections. As in the classical case, we find that the runtime scales exponentially with the number of vertices. However, we provide numerical evidence that the average runtime of the problem-specific part of the algorithm scales polynomially in the number of edges.