Sum rule for non-adiabatic geometric phases

TL;DR

This paper establishes sum rules for non-adiabatic geometric phases, revealing limitations on geometric quantum gates and impacting qudit computation strategies.

Contribution

It introduces sum rules for geometric phases in non-adiabatic evolution, extending concepts from Berry monopoles to dynamic scenarios.

Findings

Sum rules constrain geometric phase contributions in quantum systems.

Implications for the design of geometric quantum gates.

Limitations on purely geometric qudit operations.

Abstract

Berry monopoles always cancel when summing over a complete set of energy eigenstates. We demonstrate that analogous sum rules exist for geometric phases and their underlying 2-forms in non-adiabatic evolution. Our result has implications for qudit computation as it limits the types of gates that can be implemented by purely geometric means.