Quantum Search with the Signless Laplacian

TL;DR

This paper investigates quantum spatial search using the signless Laplacian on bipartite graphs, demonstrating its potential to outperform traditional Laplacian and adjacency matrix approaches in certain regimes.

Contribution

It introduces the signless Laplacian as a new operator for quantum walks in spatial search, analyzing its effectiveness on bipartite graphs and identifying regimes where it is optimal.

Findings

Signless Laplacian-based quantum walk can efficiently find marked vertices in bipartite graphs.

A specific initial state guarantees perfect evolution to marked vertices.

In some regimes, signless Laplacian outperforms Laplacian and adjacency matrix methods.

Abstract

Continuous-time quantum walks are typically effected by either the discrete Laplacian or the adjacency matrix. In this paper, we explore a third option: the signless Laplacian, which has applications in algebraic graph theory and may arise in layered antiferromagnetic materials. We explore spatial search on the complete bipartite graph, which is generally irregular and breaks the equivalence of the three quantum walks for regular graphs, and where the search oracle breaks the equivalence of the Laplacian and signless Laplacian quantum walks on bipartite graphs without the oracle. We prove that a uniform superposition over all the vertices of the graph partially evolves to the marked vertices in one partite set, with the choice of set depending on the jumping rate of the quantum walk. We boost this success probability to 1 by proving that a particular non-uniform initial state completely evolves to the marked vertices in one partite set, again depending on the jumping rate. For some parameter regimes, the signless Laplacian yields the fastest search algorithm of the three, suggesting that it could be a new tool for developing faster quantum algorithms.