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We consider the Quantum Natural Gradient Descent (QNGD) scheme which wasrecently proposed to train variational quantum algorithms. QNGD is SteepestGradient Descent (SGD) operating on the complex projective space equipped withthe Fubini-Study metric. Here we present an adaptive implementation of QNGDbased on Armijo's rule, which is an efficient backtracking line search thatenjoys a proven convergence. The proposed algorithm is tested using noisysimulators on three different models with various initializations. Our resultsshow that Adaptive QNGD dynamically adapts the step size and consistentlyoutperforms the original QNGD, which requires knowledge of optimal step size to{perform competitively}. In addition, we show that the additional complexityinvolved in performing the line search in Adaptive QNGD is minimal, ensuringthe gains provided by the proposed adaptive strategy dominates any increase incomplexity. Additionally, our benchmarking demonstrates that a simple SGDalgorithm (implemented in the Euclidean space) equipped with the adaptivescheme above, can yield performances similar to the QNGD scheme with optimalstep size.  Our results are yet another confirmation of the importance of differentialgeometry in variational quantum computations. As a matter of fact, we foreseeadvanced mathematics to play a prominent role in the NISQ era in guiding thedesign of faster and more efficient algorithms.

Quantum Natural Gradient with Efficient Backtracking Line Search