The Hamiltonians generating one-dimensional discrete-time quantum walks

TL;DR

This paper derives explicit Hamiltonians for one-dimensional discrete-time quantum walks, showing how they relate to continuous-time quantum walks in a certain limit, thus bridging two quantum walk models.

Contribution

It provides an explicit formula for Hamiltonians generating 1D discrete-time quantum walks and connects them to continuous-time quantum walks via a limiting process.

Findings

Explicit Hamiltonian formula for 1D discrete-time quantum walks

Demonstration of continuous-time quantum walks as a limit of discrete-time models

Connection to a problem proposed by Ambainis

Abstract

An explicit formula of the Hamiltonians generating one-dimensional discrete-time quantum walks is given. The formula is deduced by using the algebraic structure introduced previously. The square of the Hamiltonian turns out to be an operator without, essentially, the `coin register', and hence it can be compared with the one-dimensional continuous-time quantum walk. It is shown that, under a limit with respect to a parameter, which expresses the magnitude of the diagonal components of the unitary matrix defining the discrete-time quantum walks, the one-dimensional continuous-time quantum walk is obtained from operators defined through the Hamiltonians of the one-dimensional discrete-time quantum walks. Thus, this result can be regarded, in one-dimension, as a partial answer to a problem proposed by Ambainis.