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In a previous version of this document we misinterpreted the runtime of apart of the described algorithm. Indeed, the runtime is not better than theGrover-Algorithm. We therefor withdraw this work.  We present a novel algorithmic approach to find a proper vertex colouring ofa graph with $d$ colours, if it exists. We associate a $d$-dimensional quantumsystem with each vertex and the initial state is a mixture of all possiblecolourings, from which we obtain a random proper colouring of the graph bymeasurements. The non-deterministic nature of the quantum measurement istackled by a reset operation, which can revert the effect of unwantedprojections. As in the classical case, we find that the runtime scalesexponentially with the number of vertices. However, we provide numericalevidence that the average runtime of the problem-specific part of the algorithmscales polynomially in the number of edges.

Withdrawn: A Measurement-based Algorithm for Graph Colouring