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The two major discrete time formulations for quantum walks, coined andscattering, are unitarily equivalent for arbitrary position dependenttransition amplitudes and any topology (PRA {\bf 80}, 052301 (2009)). Althoughthe proof explicit describes the mapping obtention, its high technicality mayhinder relevant physical aspects involved in the equivalence. Discussingconcrete examples -- the most general constructions for the line, square andhoneycomb lattices -- here we unveil the similarities and differences of thesetwo versions of quantum walks. We moreover show how to derive the dynamics ofone from the other by means of proper projections. We perform calculations fordifferent probability amplitudes like, Hadamard, Grover, Discrete FourierTransform and the uncommon in the area (but interesting) Discrete HartleyTransform, comparing the evolutions. Our study illustrates the modelsinterplay, an important issue for implementations and applications of suchsystems.

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