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Bulletin of the American Physical Society

Using the graph isomorphism problem to probe differences between discrete- and continuous-time quantum random walks

Kenneth Rudinger, John King Gamble, Eric Bach, Mark G. Friesen, Robert Joynt, S. N. Coppersmith

Published Mar 18, 2013• 0 citations
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Comparing Algorithms for Graph Isomorphism Using Discrete- and Continuous-Time Quantum Random Walks

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Berry and Wang [Phys. Rev. A {\bf 83}, 042317 (2011)] show numerically that a discrete-time quantum random walk of two noninteracting particles is able to distinguish some non-isomorphic strongly regular graphs from the same family. Here we analytically demonstrate how it is possible for these walks to distinguish such graphs, while continuous-time quantum walks of two noninteracting particles cannot. We show analytically and numerically that even single-particle discrete-time quantum random walks can distinguish some strongly regular graphs, though not as many as two-particle noninteracting discrete-time walks. Additionally, we demonstrate how, given the same quantum random walk, subtle differences in the graph certificate construction algorithm can nontrivially impact the walk's distinguishing power. We also show that no continuous-time walk of a fixed number of particles can distinguish all strongly regular graphs when used in conjunction with any of the graph certificates we consider. We extend this constraint to discrete-time walks of fixed numbers of noninteracting particles for one kind of graph certificate; it remains an open question as to whether or not this constraint applies to the other graph certificates we consider.

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Given the extensive application of classical random walks to classical algorithms in a variety of fields, their quantum analogue in quantum walks is expected to provide a fruitful source of quantum algorithms. So far, however, such algorithms have been scarce. In this work, we enumerate some important differences between quantum and classical walks, leading to their markedly different properties. We show that for many practical purposes, the implementation of quantum walks can be efficiently achieved using a classical computer. We then develop both classical and quantum graph isomorphism algorithms based on discrete-time quantum walks. We show that they are effective in identifying isomorphism classes of large databases of graphs, in particular groups of strongly regular graphs. We consider this approach to represent a promising candidate for an efficient solution to the graph isomorphism problem, and believe that similar methods employing quantum walks, or derivatives of these walks, may prove beneficial in constructing other algorithms for a variety of purposes.

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An enhanced classical approach to graph isomorphism using continuous-time quantum walk

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Studies on graph isomorphism play an important role in graph research, and graph isomorphism algorithms have a wide range of applications in image matching, pattern recognition, computer vision, biochemistry and other fields. Previous research demonstrated that involving discrete-time quantum walk in the graph isomorphism algorithm could achieve complexity O ( N 7 ) for general graphs, since quantum walk could be utilized as a new toolbox for solving graph problems. We develop an enhanced classical approach to graph isomorphism using continuous-time quantum walk, which has lower complexity O ( N 5 ) and can effectively distinguish the graphs that are generally considered difficult. In addition, we define a graph similarity measure based on the proposed algorithm, which can be used for graph isomorphism and graph clustering. In the experiment, we test a wide variety of classes of graphs; the results show that the algorithm has a wide range of applications rather than being limited to a specific type of graph.

Journal of Physics A Mathematical and Theoretical•Jan 4, 2012•23 citations
Berry and Wang [Phys. Rev. A {\bf 83}, 042317 (2011)] show numerically that a discrete-time quantum random walk of two noninteracting particles is able to distinguish some non-isomorphic strongly regular graphs from the same family. Here we analytically demonstrate how it is possible for these walks to distinguish such graphs, while continuous-time quantum walks of two noninteracting particles cannot. We show analytically and numerically that even single-particle discrete-time quantum random walks can distinguish some strongly regular graphs, though not as many as two-particle noninteracting discrete-time walks. Additionally, we demonstrate how, given the same quantum random walk, subtle differences in the graph certificate construction algorithm can nontrivially impact the walk's distinguishing power. We also show that no continuous-time walk of a fixed number of particles can distinguish all strongly regular graphs when used in conjunction with any of the graph certificates we consider. We extend this constraint to discrete-time walks of fixed numbers of noninteracting particles for one kind of graph certificate; it remains an open question as to whether or not this constraint applies to the other graph certificates we consider.
Given the extensive application of classical random walks to classical algorithms in a variety of fields, their quantum analogue in quantum walks is expected to provide a fruitful source of quantum algorithms. So far, however, such algorithms have been scarce. In this work, we enumerate some important differences between quantum and classical walks, leading to their markedly different properties. We show that for many practical purposes, the implementation of quantum walks can be efficiently achieved using a classical computer. We then develop both classical and quantum graph isomorphism algorithms based on discrete-time quantum walks. We show that they are effective in identifying isomorphism classes of large databases of graphs, in particular groups of strongly regular graphs. We consider this approach to represent a promising candidate for an efficient solution to the graph isomorphism problem, and believe that similar methods employing quantum walks, or derivatives of these walks, may prove beneficial in constructing other algorithms for a variety of purposes.
Studies on graph isomorphism play an important role in graph research, and graph isomorphism algorithms have a wide range of applications in image matching, pattern recognition, computer vision, biochemistry and other fields. Previous research demonstrated that involving discrete-time quantum walk in the graph isomorphism algorithm could achieve complexity O ( N 7 ) for general graphs, since quantum walk could be utilized as a new toolbox for solving graph problems. We develop an enhanced classical approach to graph isomorphism using continuous-time quantum walk, which has lower complexity O ( N 5 ) and can effectively distinguish the graphs that are generally considered difficult. In addition, we define a graph similarity measure based on the proposed algorithm, which can be used for graph isomorphism and graph clustering. In the experiment, we test a wide variety of classes of graphs; the results show that the algorithm has a wide range of applications rather than being limited to a specific type of graph.