  Quantum Approximate Optimization Algorithm (QAOA) is a hybrid algorithm whose
control parameters are classically optimized. In addition to the variational
parameters, the right choice of hyperparameter is crucial for improving the
performance of any optimization model. Control depth, or the number of
variational parameters, is considered as the most important hyperparameter for
QAOA. In this paper we investigate the control depth selection with an
automatic algorithm based on proximal gradient descent. The performances of the
automatic algorithm are demonstrated on 7-node and 10-node Max-Cut problems,
which show that the control depth can be significantly reduced during the
iteration while achieving an sufficient level of optimization accuracy. With
theoretical convergence guarantee, the proposed algorithm can be used as an
efficient tool for choosing the appropriate control depth as a replacement of
random search or empirical rules. Moreover, the reduction of control depth will
induce a significant reduction in the number of quantum gates in circuit, which
improves the applicability of QAOA on Noisy Intermediate-scale Quantum (NISQ)
devices.
