An Effective Hamiltonian Approach to Quantum Random Walk

TL;DR

This paper introduces an effective Hamiltonian framework for discrete-time quantum random walks, generalizing to higher dimensions, and compares additive and multiplicative evolution operators through computational studies on various lattices.

Contribution

It presents a novel Hamiltonian formulation for quantum walks, extending the approach to higher dimensions and analyzing the differences between additive and multiplicative evolution operators.

Findings

The Hamiltonian approach aligns with standard methods in 1D.

In higher dimensions, the evolution operator is additive, not multiplicative.

On Graphene lattices, the additive approach is preferred.

Abstract

In this article we present an effective Hamiltonian approach for Discrete Time Quantum Random Walk. A form of the Hamiltonian for one dimensional quantum walk has been prescribed, utilizing the fact that Hamiltonians are the generators of time translations. Then an attempt has been made to generalize the techniques to higher dimensions. We find that the Hamiltonian can be written as the sum of a Weyl Hamiltonian and a Dirac comb potential. The time evolution operator obtained from this prescribed Hamiltonian is in complete agreement with that of the standard approach. But in higher dimension we find that the time evolution operator is additive, instead of being multiplicative \cite{Chandrasekhar:2013SREP08229}. We showed that in case of two-step walk, effectively the time evolution operator can have multiplicative form. In case of a square lattice, quantum walk has been studied computationally for different coins and the results for both the additive and the multiplicative approaches have been compared. Using the Graphene Hamiltonian the walk has been studied on a Graphene lattice and we conclude the preference of additive approach over the multiplicative one.