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arXiv:1705.11036 · New Journal of Physics

Quantum walk on a chimera graph

Shu Xu, Xiangxiang Sun, Jizhou Wu, Weiwei Zhang, Nigum Arshed, Barry C. Sanders

Published Mar 15, 2018• 5 citations
arXivHTMLPDF

Abstract

We analyse a continuous-time quantum walk on a chimera graph, which is a graph of choice for designing quantum annealers, and we discover beautiful quantum walk features such as localization that starkly distinguishes classical from quantum behaviour. Motivated by technological thrusts, we study continuous-time quantum walk on enhanced variants of the chimera graph and on diminished chimera graph with a random removal of vertices. We explain the quantum walk by constructing a generating set for a suitable subgroup of graph isomorphisms and corresponding symmetry operators that commute with the quantum walk Hamiltonian; the Hamiltonian and these symmetry operators provide a complete set of labels for the spectrum and the stationary states. Our quantum walk characterization of the chimera graph and its variants yields valuable insights into graphs used for designing quantum-annealers.

DOI
10.1088/1367-2630/aab701

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We analyse a continuous-time quantum walk on a chimera graph, which is a graph of choice for designing quantum annealers, and we discover beautiful quantum walk features such as localization that starkly distinguishes classical from quantum behaviour. Motivated by technological thrusts, we study continuous-time quantum walk on enhanced variants of the chimera graph and on diminished chimera graph with a random removal of vertices. We explain the quantum walk by constructing a generating set for a suitable subgroup of graph isomorphisms and corresponding symmetry operators that commute with the quantum walk Hamiltonian; the Hamiltonian and these symmetry operators provide a complete set of labels for the spectrum and the stationary states. Our quantum walk characterization of the chimera graph and its variants yields valuable insights into graphs used for designing quantum-annealers.
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Motivated by the immense success of random walk and Markov chain methods in the design of classical algorithms, we consider_quantum_ walks on graphs. We analyse in detail the behaviour of unbiased quantum walk on the line, with the example of a typical walk, the ``Hadamard walk''. We show that after t time steps, the probability distribution on the line induced by the Hadamard walk is almost uniformly distributed over the interval [-t/sqrt(2),t/sqrt(2)]. This implies that the same walk defined on the circle mixes in_linear_ time. This is in direct contrast with the quadratic mixing time for the corresponding classical walk. We conclude by indicating how our techniques may be applied to more general graphs.
Graph isomorphism is an essential problem in graph theory and widely used in a variety of applications. With graph size increasing, graph isomorphism become difficult for classical computation since there is no polynomial algorithm for general graphs yet. Considering the advantages of quantum computation over classical computation in various tasks, quantum graph isomorphism algorithms have been studied to explore the potentials of quantum computation in graph isomorphism. In this work, we propose a graph isomorphism algorithm based on continuous-time quantum walks, where we construct the graph certificates from the output probabilities of continuous-time quantum walks on the tested graphs with added self-loops to graph vertices, and distinguish graphs by comparing the graph certificates. Numerous tests on diverse graph classes demonstrate the effectiveness of the proposed GI algorithm, which is not limited to a specific topological structure graph. Moreover, we test the proposed algorithm on classes of strongly regular graphs, and the results show that the distinguishing ability of the algorithm can be improved with the increasement of the added self-loops to distinct vertices.
We set the ground for a theory of quantum walks on graphs- the generalization of random walks on finite graphs to the quantum world. Such quantum walks do not converge to any stationary distribution, as they are unitary and reversible. However, by suitably relaxing the definition, we can obtain a measure of how fast the quantum walk spreads or how confined the quantum walk stays in a small neighborhood. We give definitions of mixing time, filling time, dispersion time. We show that in all these measures, the quantum walk on the cycle is almost quadratically faster then its classical correspondent. On the other hand, we give a lower bound on the possible speed up by quantum walks for general graphs, showing that quantum walks can be at most polynomially faster than their classical counterparts.
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