Search by Lackadaisical Quantum Walk with Symmetry Breaking

TL;DR

This paper demonstrates that lackadaisical quantum walks with symmetry-breaking self-loops can maintain search efficiency on vertex-transitive graphs, even when the self-loop weights break graph symmetries.

Contribution

It introduces the idea that symmetry-breaking self-loops, with weights at the marked vertex, do not impair the quantum walk's search speedup, expanding the understanding of quantum walk robustness.

Findings

Symmetry-breaking self-loops preserve search speedup.

Only the marked vertex's self-loop weight is crucial.

Random small weights at other vertices do not affect performance.

Abstract

The lackadaisical quantum walk is a lazy version of a discrete-time, coined quantum walk, where each vertex has a weighted self-loop that permits the walker to stay put. They have been used to speed up spatial search on a variety of graphs, including periodic lattices, strongly regular graphs, Johnson graphs, and the hypercube. In these prior works, the weights of the self-loops preserved the symmetries of the graphs. In this paper, we show that the self-loops can break all the symmetries of vertex-transitive graphs while providing the same computational speedups. Only the weight of the self-loop at the marked vertex matters, and the remaining self-loop weights can be chosen randomly, as long as they are small compared to the degree of the graph.