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arXiv:quant-ph/0404136 · Journal of Physics A Mathematical and General

An approximation to couplings on graphs

Taksu Cheon, Pavel Exner

Published Jul 8, 2004• 39 citations
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Abstract

We discuss a general parametrization for vertices of quantum graphs and show, in particular, how the δ' s and δ' couplings at an n -edge vertex can be approximated by means of n + 1 couplings of the δ type provided the latter are properly scaled.
DOI
10.1088/0305-4470/37/29/l01

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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 ddd colours, if it exists. We associate a ddd-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.