Information geometry of nonmonotonic quantum natural gradient

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.