Complex-phase extensions of the Szegedy quantum walk on graphs

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.