Information geometry of nonmonotonic quantum natural gradient

TL;DR

This paper explores the properties of nonmonotonic quantum natural gradient, demonstrating its potential for faster convergence in quantum optimization by analyzing quantum Fisher metrics and providing numerical comparisons.

Contribution

It introduces a detailed analysis of nonmonotonic QNG, extending quantum Fisher metric theory to non-full-rank states and exploring geometry design via Petz functions.

Findings

Nonmonotonic QNG can achieve faster convergence than traditional QNG.

Quantum Fisher metrics can be designed using Petz functions.

Numerical simulations show improved performance in quantum circuit learning.

Abstract

Natural gradient is an advanced optimization method based on information geometry, where the Fisher metric plays a crucial role. Its quantum counterpart, known as quantum natural gradient (QNG), employs the symmetric logarithmic derivative (SLD) metric, one of the quantum Fisher metrics. While quantization in physics is typically well-defined via the canonical commutation relations, the quantization of information-theoretic quantities introduces inherent arbitrariness. To resolve this ambiguity, monotonicity has been used as a guiding principle for constructing geometries in physics, as it aligns with physical intuition. Recently, a variant of QNG, which we refer to as nonmonotonic QNG in this paper, was proposed by relaxing the monotonicity condition. It was shown to achieve faster convergence compared to conventional QNG. In this paper, we investigate the properties of nonmonotonic QNG. To ensure the paper is self-contained, we first demonstrate that the SLD metric is locally optimal under the monotonicity condition and that non-monotone quantum Fisher metrics can lead to faster convergence in QNG. Previous studies primarily relied on a specific type of quantum divergence and assumed that density operators are full-rank. Here, we explicitly consider an alternative quantum divergence and extend the analysis to non-full-rank cases. Additionally, we explore how geometries can be designed using Petz functions, given that quantum Fisher metrics are characterized through them. Finally, we present numerical simulations comparing different quantum Fisher metrics in the context of parameter estimation problems in quantum circuit learning.