Search on vertex-transitive graphs by lackadaisical quantum walk

Mason L. Rhodes, Thomas G. Wong

Research output: Contribution to journalArticle

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.

Original languageEnglish (US)
Article number334
JournalQuantum Information Processing
Volume19
Issue number9
DOIs
StatePublished - Aug 1 2020

All Science Journal Classification (ASJC) codes

  • Electronic, Optical and Magnetic Materials
  • Statistical and Nonlinear Physics
  • Theoretical Computer Science
  • Signal Processing
  • Modeling and Simulation
  • Electrical and Electronic Engineering

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