Unveiling and exemplifying the unitary equivalence of discrete time quantum walk models

TL;DR

This paper demonstrates the unitary equivalence of two major discrete-time quantum walk models, providing concrete examples and methods to derive one from the other, which is crucial for practical implementations.

Contribution

It explicitly compares coined and scattering quantum walks on various lattices, illustrating their equivalence and how to transform between them using projections.

Findings

Coined and scattering quantum walks are unitarily equivalent for arbitrary topologies.

Explicit methods to derive one model's dynamics from the other are provided.

Comparative analysis of different probability amplitudes shows their effects on walk evolution.

Abstract

The two major discrete time formulations for quantum walks, coined and scattering, are unitarily equivalent for arbitrary position dependent transition amplitudes and any topology (PRA {\bf 80}, 052301 (2009)). Although the proof explicit describes the mapping obtention, its high technicality may hinder relevant physical aspects involved in the equivalence. Discussing concrete examples -- the most general constructions for the line, square and honeycomb lattices -- here we unveil the similarities and differences of these two versions of quantum walks. We moreover show how to derive the dynamics of one from the other by means of proper projections. We perform calculations for different probability amplitudes like, Hadamard, Grover, Discrete Fourier Transform and the uncommon in the area (but interesting) Discrete Hartley Transform, comparing the evolutions. Our study illustrates the models interplay, an important issue for implementations and applications of such systems.