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arXiv:1010.0764 · Physical Review A

Quantum-walk-based search and centrality

Scott M. Berry, Jingbo B. Wang

Published Oct 26, 2010• 80 citations
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Abstract

We study the discrete-time quantum-walk-based search for a marked vertex on a graph. By considering various structures in which not all vertices are equivalent, we investigate the relationship between the successful search probability and the position of the marked vertex, in particular, its centrality. We find that the maximum value of the search probability does not necessarily increase as the marked vertex becomes more central, and we investigate an interesting relationship between the frequency of the successful search probability and the centrality of the marked vertex.

DOI
10.1103/physreva.82.042333

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Abstract The spatial search problem aims to find a marked vertex of a finite graph using a dynamic with two constraints: (1) the walker has no compass and (2) the walker can check whether a vertex is marked only after reaching it. This problem is a generalization of unsorted database search and has many applications to algorithms. Classical algorithms that solve the spatial search problem are based on random walks and the computational complexity is determined by the hitting time. On the other hand, quantum algorithms are based on quantum walks and the computational complexity is determined not only by the number of steps to reach a marked vertex, but also by the success probability, since we need to perform a measurement at the end of the algorithm to determine the walker’s position. In this work, we address the spatial search problem on Johnson graphs using the coined quantum walk model. Since Johnson graphs are vertex- and distance-transitive, we have found an invariant subspace of the Hilbert space, which aids in the calculation of the computational complexity. We have shown that, for every fixed diameter, the asymptotic success probability is 1/2 after taking π N / ( 2 2 ) steps, where N is the number of vertices of the Johnson graph.
© 2014 IOP Publishing Ltd. In this paper we present a lemma, which helps us to establish a link between the distribution of success probabilities from quantum walk based search and the symmetries of the underlying graphs. With the aid of the lemma, we identified certain graph structures of which the quantum walk based search provides high success probabilities at the marked vertices. We also observed that many graph structures and their vertices can be classified according to their structural equivalence using the search probabilities provided by quantum walks, although this method cannot resolve all non-equivalent vertices for strongly regular graphs.
Continuous-time quantum walks are natural tools for spatial search, where one searches for a marked vertex in a graph. Sometimes the structure of the graph causes the walker to get trapped, such that the probability of finding the marked vertex is limited. We give an example with two linked cliques, proving that the captive probability can be liberated by increasing the weights of the links. This allows the search to succeed with probability 1 without increasing the energy scaling of the algorithm. Further increasing the weights, however, slows the runtime, so the optimal search requires weights that are neither too weak nor too strong.