Particle-Number Threshold for Non-Abelian Geometric Phases

TL;DR

This paper introduces a particle-number threshold (PNT) to determine the minimum number of particles needed for a quantum system to fully realize non-Abelian geometric phases, aiding in resource assessment for holonomic quantum computing.

Contribution

The paper defines and analyzes a particle-number threshold (PNT) that quantifies the resource requirements for generating non-Abelian geometric phases in quantum systems.

Findings

PNT provides a clear criterion for the minimal particle number needed.

Benchmarking on bosonic systems demonstrates the PNT's practical relevance.

The approach helps evaluate the resource demands of holonomic quantum computers.

Abstract

When a quantum state traverses a path, while being under the influence of a gauge potential, it acquires a geometric phase that is often more than just a scalar quantity. The variety of unitary transformations that can be realised by this form of parallel transport depends crucially on the number of particles involved in the evolution. Here, we introduce a particle-number threshold (PNT) that assesses a system's capabilities to perform purely geometric manipulations of quantum states. This threshold gives the minimal number of particles necessary to fully exploit a system's potential to generate non-Abelian geometric phases. Therefore, the PNT might be useful for evaluating the resource demands of a holonomic quantum computer. We benchmark our findings on bosonic systems relevant to linear and nonlinear quantum optics.