Laplacian versus Adjacency Matrix in Quantum Walk Search

TL;DR

This paper compares the effects of using Laplacian versus adjacency matrices in continuous-time quantum walk algorithms, revealing significant differences in performance and behavior on non-regular graphs like bipartite graphs.

Contribution

It provides a detailed analysis of how the choice between Laplacian and adjacency matrices impacts quantum walk search algorithms on non-regular graphs.

Findings

Laplacian and adjacency matrix walks differ in required jumping rate.

They have different runtimes and sampling behaviors.

The initial state choice is affected by the matrix used.

Abstract

A quantum particle evolving by Schr\"odinger's equation contains, from the kinetic energy of the particle, a term in its Hamiltonian proportional to Laplace's operator. In discrete space, this is replaced by the discrete or graph Laplacian, which gives rise to a continuous-time quantum walk. Besides this natural definition, some quantum walk algorithms instead use the adjacency matrix to effect the walk. While this is equivalent to the Laplacian for regular graphs, it is different for non-regular graphs, and is thus an inequivalent quantum walk. We algorithmically explore this distinction by analyzing search on the complete bipartite graph with multiple marked vertices, using both the Laplacian and adjacency matrix. The two walks differ qualitatively and quantitatively in their required jumping rate, runtime, sampling of marked vertices, and in what constitutes a natural initial state. Thus the choice of the Laplacian or adjacency matrix to effect the walk has important algorithmic consequences.