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

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.