Kinematics and Dynamics of Quantum Walks in terms of Systems of Imprimitivity

TL;DR

This paper develops systems of imprimitivity for quantum walks, providing geometric constructions and exploring their applications in different symmetry contexts, including de Sitter space, spacetime symmetries, and graphs.

Contribution

It introduces a geometric framework for systems of imprimitivity in quantum walks, extending their application to various symmetry groups and graph structures.

Findings

Constructed SI for quantum walks on de Sitter space.

Analyzed SI on orbits of stabilizer subgroups in spacetime.

Applied SI constructions to automorphisms on distant-transitive graphs.

Abstract

We build systems of imprimitivity (SI) in the context of quantum walks and provide geometric constructions for their configuration space. We consider three systems, an evolution of unitaries from the group SO3 on a low dimensional de Sitter space where the walk happens on the dual of SO3, standard quantum walk whose SI live on the orbits of stabilizer subgroups (little groups) of semidirect products describing the symmetries of 1+1 spacetime, and automorphisms (walks are specific automorphisms) on distant-transitive graphs as application of the constructions.