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Search on Vertex-Transitive Graphs by Lackadaisical Quantum Walk

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.