One of the most important properties of classical neural networks is the clustering of local minima of the network near the global minimum, enabling efficient training. This has been observed not only numerically, but also has begun to be analytically understood through the lens of random matrix theory. Inspired by these results in classical machine learning, we show that a certain randomized class of variational quantum algorithms can be mapped to Wishart random fields on the hypertorus. Then, using the statistical properties of such random processes, we analytically find the expected distribution of critical points. Unlike the case for deep neural networks, we show the existence of a transition in the quality of local minima at a number of parameters exponentially large in the problem size. Below this transition, all local minima are concentrated far from the global minimum; above, all local minima are concentrated near the global minimum. This is consistent with previously observed numerical results on the landscape behavior of Hamiltonian agnostic variational quantum algorithms. We give a heuristic explanation as to why ansatzes that depend on the problem Hamiltonian might not suffer from these scaling issues. We also verify that our analytic results hold experimentally even at modest system sizes.
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