Quantum Natural Gradient with Geodesic Corrections for Small Shallow Quantum Circuits

TL;DR

This paper introduces an enhanced quantum natural gradient method with geodesic corrections, improving optimization efficiency in variational quantum algorithms by leveraging geometric insights and higher-order integrators.

Contribution

We extend the quantum natural gradient method by incorporating geodesic corrections and higher-order integrators, enabling more efficient quantum optimization.

Findings

QNGGC significantly improves convergence rates over standard QNG.

The method efficiently computes Christoffel symbols using the parameter-shift rule.

Geodesic corrections enhance the optimization process in variational quantum algorithms.

Abstract

The Quantum Natural Gradient (QNG) method enhances optimization in variational quantum algorithms (VQAs) by incorporating geometric insights from the quantum state space through the Fubini-Study metric. In this work, we extend QNG by introducing higher-order integrators and geodesic corrections using the Riemannian Euler update rule and geodesic equations, deriving an updated rule for the Quantum Natural Gradient with Geodesic Correction (QNGGC). We also develop an efficient method for computing the Christoffel symbols necessary for these corrections, leveraging the parameter-shift rule to enable direct measurement from quantum circuits. Through theoretical analysis and practical examples, we demonstrate that QNGGC significantly improves convergence rates over standard QNG, highlighting the benefits of integrating geodesic corrections into quantum optimization processes. Our approach paves the way for more efficient quantum algorithms, leveraging the advantages of geometric methods.