Complex-Phase Extensions of Szegedy Quantum Walk on Graphs

TL;DR

This paper introduces a graph-phased Szegedy quantum walk model with link phases and local phase rotations, enhancing quantum algorithm capabilities and broadening the equivalence with coined quantum walks.

Contribution

It presents a novel graph-phased Szegedy quantum walk framework, including circuit adaptations, phase marking techniques, and an improved classical simulation algorithm.

Findings

Enhanced quantum search efficiency using local APR phases.

Broader class of coin operators compatible with the model.

Improved classical simulation algorithm for quantum walks.

Abstract

This work introduces a graph-phased Szegedy's quantum walk, which incorporates link phases and local arbitrary phase rotations (APR), unlocking new possibilities for quantum algorithm efficiency. We demonstrate how to adapt quantum circuits to these advancements, allowing phase patterns that ensure computational practicality. The graph-phased model broadens the known equivalence between coined quantum walks and Szegedy's model, accommodating a wider array of coin operators. Through illustrative examples, we reveal intriguing disparities between classical and quantum interpretations of walk dynamics. Remarkably, local APR phases emerge as powerful tools for marking graph nodes, optimizing quantum searches without altering graph structure. We further explore the surprising nuances between single and double operator approaches, highlighting a greater range of compatible coins with the latter. To facilitate these advancements, we present an improved classical simulation algorithm, which operates with superior efficiency. This study not only refines quantum walk methodologies but also paves the way for future explorations, including potential applications in quantum search and PageRank algorithms. Our findings illuminate the path towards more versatile and powerful quantum computing paradigms.