Classification of unitary operators by local generatability

TL;DR

This paper classifies noninteracting unitary operators in all dimensions based on their local generatability, revealing that equivalence up to local generation aligns with homotopy equivalence across all symmetry classes.

Contribution

It introduces a classification framework for unitary operators based on local generatability and establishes their equivalence with homotopy classes in all symmetry classes and dimensions.

Findings

Equivalence up to local generation equals homotopy equivalence.

Classification applies to all Altland-Zirnbauer symmetry classes.

Provides a comprehensive framework for noninteracting unitaries.

Abstract

Periodically driven (Floquet) systems can exhibit possibilities beyond what can be obtained in equilibrium. Both in Floquet systems and in the related problems of discrete-time quantum walks and quantum cellular automata, a basic distinction arises among unitary time evolution operators: while all physical operators are local, not all are locally generated (i.e., generated by some local Hamiltonian). In this paper, we define the notion of equivalence up to a locally generated unitary in all Altland-Zirnbauer symmetry classes. We then classify noninteracting unitaries in all dimensions on this basis by showing that equivalence up to a locally generated unitary is identical to homotopy equivalence.