Equivalence between discrete quantum walk models in arbitrary topologies

TL;DR

This paper proves the unitary equivalence between coin and scattering models of discrete quantum walks on arbitrary graphs, providing a general construction and methods to translate probabilities between models.

Contribution

It establishes a comprehensive framework demonstrating the equivalence of coin and scattering quantum walk models on any graph topology.

Findings

Constructive proof of unitary equivalence for arbitrary topologies

Method to derive probabilities of one model from the other

Generalized construction for position-dependent transition amplitudes

Abstract

Coin and scattering are the two major formulations for discrete quantum walks models, each believed to have its own advantages in different applications. Although they are related in some cases, it was an open question their equivalence in arbitrary topologies. Here we present a general construction for the two models for any graph and also for position dependent transition amplitudes. We then prove constructively their unitary equivalence. Defining appropriate projector operators, we moreover show how to obtain the probabilities for one model from the evolution of the other.