Parameterization of Translation-Invariant Two-Dimensional Two-State Quantum Walks

Abstract

This study investigates the unitary equivalence classes of translationinvariant two-dimensional two-state quantum walks. We show that unitary equivalence classes of such quantum walks are essentially parameterized by two real parameters.

Unitary equivalence classes of split-step quantum walks

“…Then, we only need to study representatives of unitary equivalence classes to know the above properties. This helps studies of SSQW, because we can reduce the parameters which are needed to define quantum walks [5,6,9,11,12,13].…”

Parameterization of quantum walks on cycles

“…In the previous papers [7][8][9], we considered unitary equivalence classes of one-dimensional and two-dimensional quantum walks. Unitary equivalence classes of translation-invariant one-dimensional quantum walks were also investigated in [3].…”