Unitary equivalence of quantum walks

• We have found unitary equivalent classes in coined quantum walks.
• A single parameter family of coin operators is sufficient to realize all simple one-dimensional quantum walks.
• Electric quantum walks are unitarily equivalent to time dependent quantum walks.

A simple   coined quantum walk in one dimension can be characterized by a SU(2)SU(2) operator with three parameters which represents the coin toss. However, different such coin toss operators lead to equivalent dynamics of the quantum walker. In this manuscript we present the unitary equivalence classes of quantum walks and show that all the nonequivalent quantum walks can be distinguished by a single parameter. Moreover, we argue that the electric quantum walks are equivalent to quantum walks with time dependent coin toss operator.