Topological sum rule for geometric phases of quantum gates

TL;DR

This paper derives a topological sum rule linking geometric phases of two-qubit quantum gates to the Hamiltonian's winding number, revealing that nontrivial topology is essential for entanglement generation.

Contribution

It introduces a topological sum rule connecting geometric phases and Hamiltonian topology, providing a measurable way to distinguish topological classes of quantum gates.

Findings

The sum rule relates geometric phases to the Hamiltonian's winding number.

Different topological classes of gates distribute phases differently, measurable via concurrence.

Nontrivial topology is necessary for entanglement generation.

Abstract

We establish a topological sum rule, $\nu_U = \frac{1}{2\pi}\sum_n\gamma_n = 2m\nu_H$, connecting the geometric phases accumulated by a two-qubit system over a complete basis of initial states to the winding number $\nu_H$ classifying its Hamiltonian. Implementations of the same gate from different topological classes must distribute these phases differently, making their distinction measurable through the Wootters concurrence. As a corollary, nontrivial topology is a necessary condition for entanglement: only Hamiltonians with access to $\nu_H \neq 0$ can generate it.