Connection Between Continuous and Discrete Time Quantum Walks on d-Dimensional Lattices; Extensions to General Graphs

TL;DR

This paper demonstrates how continuous-time quantum walks on d-dimensional lattices can be derived as limits of discrete-time quantum walks and extends this relationship to general graphs, broadening the understanding of quantum walk dynamics.

Contribution

It generalizes the connection between discrete and continuous quantum walks from the infinite line to d-dimensional lattices and arbitrary graphs, identifying the continuous dynamics as limits of discrete steps.

Findings

Continuous-time quantum walks are limits of discrete-time walks on lattices.

Extension of the limit process to general graphs.

Identification of Hamiltonians as limits of discrete dynamics.

Abstract

I obtain the dynamics of the continuous time quantum walk on a $d$-dimensional lattice, with periodic boundary conditions, as an appropriate limit of the dynamics of the discrete time quantum walk on the same lattice. This extends the main result of arXiv:quant-ph/0606050 which proved this limit for the infinite line. By highlighting the main features of the limiting procedure, I then extend it to general graphs. For a given discrete time quantum walk on a general graph, I single out the type of continuous dynamics (Hamiltonians) that can be obtained as a limit of the discrete time dynamics.