Establishing the equivalence between Szegedy's and coined quantum walks using the staggered model

TL;DR

This paper demonstrates the conditions under which coined quantum walks are equivalent to Szegedy's quantum walks, unifying two prominent models and enabling their interchangeable use in quantum algorithms.

Contribution

It establishes the precise conditions for equivalence between coined and Szegedy's quantum walks, including their application to quantum search algorithms.

Findings

Coined and Szegedy's QWs are equivalent on converted bipartite graphs.

The evolution operators of both models can be made identical under certain conditions.

Quantum search algorithms using coined QWs can be reformulated within Szegedy's framework.

Abstract

Coined Quantum Walks (QWs) are being used in many contexts with the goal of understanding quantum systems and building quantum algorithms for quantum computers. Alternative models such as Szegedy's and continuous-time QWs were proposed taking advantage of the fact that quantum theory seems to allow different quantized versions based on the same classical model, in this case, the classical random walk. In this work, we show the conditions upon which coined QWs are equivalent to Szegedy's QWs. Those QW models have in common a large class of instances, in the sense that the evolution operators are equal when we convert the graph on which the coined QW takes place into a bipartite graph on which Szegedy's QW takes place, and vice versa. We also show that the abstract search algorithm using the coined QW model can be cast into Szegedy's searching framework using bipartite graphs with sinks.