Exact simulation of coined quantum walks with the continuous-time model

TL;DR

This paper presents a method to exactly simulate coined quantum walks using continuous-time quantum walks on an expanded graph, involving dynamic edge control to replicate the coin and shift operations.

Contribution

It introduces a novel approach to produce the evolution of coined quantum walks via continuous-time models on larger, dynamically changing graphs, expanding the structural framework.

Findings

Continuous-time walk can simulate coined walk on a larger graph.

Dynamic edge control is necessary to replicate coin and shift operations.

In some cases, the simulation uses the graph Laplacian on a changing graph.

Abstract

The connection between coined and continuous-time quantum walk models has been addressed in a number of papers. In most of those studies, the continuous-time model is derived from coined quantum walks by employing dimensional reduction and taking appropriate limits. In this work, we produce the evolution of a coined quantum walk on a generic graph using a continuous-time quantum walk on a larger graph. In addition to expanding the underlying structure, we also have to switch on and off edges during the continuous-time evolution to accommodate the alternation between the shift and coin operators from the coined model. In one particular case, the connection is very natural, and the continuous-time quantum walk that simulates the coined quantum walk is driven by the graph Laplacian on the dynamically changing expanded graph.