Large gradients via correlation in random parameterized quantum circuits

TL;DR

This paper demonstrates that introducing correlations in the parameters of large random quantum circuits can prevent vanishing gradients, thereby improving the trainability of variational quantum algorithms.

Contribution

It proves that correlated gate layers in quantum circuits can circumvent vanishing gradients, enhancing optimization in large-scale variational quantum algorithms.

Findings

Correlated parameters prevent exponential decay of gradients.

Transition from vanishing to efficient gradients with increasing layers.

Optimal layer complexity matches quantum search bounds.

Abstract

Scaling of variational quantum algorithms to large problem sizes requires efficient optimization of random parameterized quantum circuits. For such circuits with uncorrelated parameters, the presence of exponentially vanishing gradients in cost function landscapes is an obstacle to optimization by gradient descent methods. In this work, we prove that reducing the dimensionality of the parameter space by utilizing circuit modules containing spatially or temporally correlated gate layers can allow one to circumvent the vanishing gradient phenomenon. Examples are drawn from random separable circuits and asymptotically optimal variational versions of Grover's algorithm based on the quantum alternating operator ansatz (QAOA). In the latter scenario, our bounds on cost function variation imply a transition between vanishing gradients and efficient trainability as the number of layers is increased toward $\mathcal{O}(2^{n/2})$, the optimal oracle complexity of quantum unstructured search.