  Variational quantum algorithms (VQAs) are promising methods that leverage
noisy quantum computers and classical computing techniques for practical
applications. In VQAs, the classical optimizers such as gradient-based
optimizers are utilized to adjust the parameters of the quantum circuit so that
the objective function is minimized. However, they often suffer from the
so-called vanishing gradient or barren plateau issue. On the other hand, the
normalized gradient descent (NGD) method, which employs the normalized gradient
vector to update the parameters, has been successfully utilized in several
optimization problems. Here, we study the performance of the NGD methods in the
optimization of VQAs for the first time. Our goal is two-fold. The first is to
examine the effectiveness of NGD and its variants for overcoming the vanishing
gradient problems. The second is to propose a new NGD that can attain the
faster convergence than the ordinary NGD. We performed numerical simulations of
these gradient-based optimizers in the context of quantum chemistry where VQAs
are used to find the ground state of a given Hamiltonian. The results show the
effective convergence property of the NGD methods in VQAs, compared to the
relevant optimizers without normalization. Moreover, we make use of some
normalized gradient vectors at the past iteration steps to propose the novel
historical NGD that has a theoretical guarantee to accelerate the convergence
speed, which is observed in the numerical experiments as well.

Quantum Physics