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The lackadaisical quantum walk is a discrete-time, coined quantum walk on agraph with a weighted self-loop at each vertex. It uses a generalized Grovercoin and the flip-flop shift, which makes it equivalent to Szegedy's quantumMarkov chain. It has been shown that a lackadaisical quantum walk can improvespatial search on the complete graph, discrete torus, cycle, and regularcomplete bipartite graph. In this paper, we observe that these are allvertex-transitive graphs, and when there is a unique marked vertex, the optimalweight of the self-loop equals the degree of the loopless graph divided by thetotal number of vertices. We propose that this holds for all vertex-transitivegraphs with a unique marked vertex. We present a number of numericalsimulations supporting this hypothesis, including search on periodic cubiclattices of arbitrary dimension, strongly regular graphs, Johnson graphs, andthe hypercube.

Search on Vertex-Transitive Graphs by Lackadaisical Quantum Walk