Search on vertex-transitive graphs by lackadaisical quantum walk

Abstract

The lackadaisical quantum walk is a discrete-time, coined quantum walk on a graph with a weighted self-loop at each vertex. It uses a generalized Grover coin and the flip-flop shift, which makes it equivalent to Szegedy’s quantum Markov chain. It has been shown that a lackadaisical quantum walk can improve spatial search on the complete graph, discrete torus, cycle, and regular complete bipartite graph. In this paper, we observe that these are all vertex-transitive graphs, and when there is a unique marked vertex, the optimal weight of the self-loop equals the degree of the loopless graph divided by the total number of vertices. We propose that this holds for all vertex-transitive graphs with a unique marked vertex. We present a number of numerical simulations supporting this hypothesis, including search on periodic cubic lattices of arbitrary dimension, strongly regular graphs, Johnson graphs, and the hypercube.

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Acknowledgements

Thanks to Peter Høyer and Zhan Yu for useful discussions. This work was partially supported by T.W.’s startup funds from Creighton University

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Correspondence to Thomas G. Wong.

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Rhodes, M.L., Wong, T.G. Search on vertex-transitive graphs by lackadaisical quantum walk. Quantum Inf Process 19, 334 (2020). https://doi.org/10.1007/s11128-020-02841-z

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Keywords

  • Quantum walk
  • Lackadaisical quantum walk
  • Quantum search
  • Spatial search
  • Vertex transitive graph