Quantum Natural Gradient with Efficient Backtracking Line Search

Abstract: We consider the Quantum Natural Gradient Descent (QNGD) scheme which was recently proposed to train variational quantum algorithms. QNGD is Steepest Gradient Descent (SGD) operating on the complex projective space equipped with the Fubini-Study metric. Here we present an adaptive implementation of QNGD based on Armijo's rule, which is an efficient backtracking line search that enjoys a proven convergence. The proposed algorithm is tested using noisy simulators on three different models with various initializations. Our results show that Adaptive QNGD dynamically adapts the step size and consistently outperforms the original QNGD, which requires knowledge of optimal step size to {perform competitively}. In addition, we show that the additional complexity involved in performing the line search in Adaptive QNGD is minimal, ensuring the gains provided by the proposed adaptive strategy dominates any increase in complexity. Additionally, our benchmarking demonstrates that a simple SGD algorithm (implemented in the Euclidean space) equipped with the adaptive scheme above, can yield performances similar to the QNGD scheme with optimal step size. Our results are yet another confirmation of the importance of differential geometry in variational quantum computations. As a matter of fact, we foresee advanced mathematics to play a prominent role in the NISQ era in guiding the design of faster and more efficient algorithms.