Connecting geometry and performance of two-qubit parameterized quantum circuits

TL;DR

This paper explores the geometric properties of two-qubit parameterized quantum circuits using principal bundles and metrics, revealing how curvature relates to entanglement and impacts optimization performance.

Contribution

It introduces a geometric framework for analyzing two-qubit PQCs, linking curvature and entanglement, and explains the effectiveness of Quantum Natural Gradient in optimization.

Findings

Ricci scalar relates to entanglement via Mannoury-Fubini-Study metric.

Quantum Natural Gradient finds high negative curvature regions early, accelerating optimization.

Curvature analysis explains performance differences between gradient methods.

Abstract

Parameterized quantum circuits (PQCs) are a central component of many variational quantum algorithms, yet there is a lack of understanding of how their parameterization impacts algorithm performance. We initiate this discussion by using principal bundles to geometrically characterize two-qubit PQCs. On the base manifold, we use the Mannoury-Fubini-Study metric to find a simple equation relating the Ricci scalar (geometry) and concurrence (entanglement). By calculating the Ricci scalar during a variational quantum eigensolver (VQE) optimization process, this offers us a new perspective to how and why Quantum Natural Gradient outperforms the standard gradient descent. We argue that the key to the Quantum Natural Gradient's superior performance is its ability to find regions of high negative curvature early in the optimization process. These regions of high negative curvature appear to be important in accelerating the optimization process.