Quantum natural gradient without monotonicity

Natural gradient (NG) is an information-geometric optimization method that plays a crucial role, especially in the estimation of parameters for machine learning models like neural networks. To apply NG to quantum systems, the quantum natural gradient (QNG) was introduced and utilized for noisy intermediate-scale devices. Additionally, a mathematically equivalent approach to QNG, known as the stochastic reconfiguration method, has been implemented to enhance the performance of quantum Monte Carlo methods. It is worth noting that these methods are based on the symmetric logarithmic derivative (SLD) metric, which is one of the monotone metrics. So far, monotonicity has been believed to be a guiding principle to construct a geometry in physics. In this paper, we propose generalized QNG by removing the condition of monotonicity. Initially, we demonstrate that monotonicity is a crucial condition for conventional QNG to be optimal. Subsequently, we provide analytical and numerical evidence showing that non-monotone QNG outperforms conventional QNG based on the SLD metric in terms of convergence speed.