The most promising class of algorithms for near-term quantum computers are variational quantum algorithms (VQAs), where a parametrised circuit of quantum gates calculates some cost function, which is then optimised to produce the solution to a problem, e.g. calculating the ground-state energy of a molecule.
Typically, the optimisation methods used in VQAs are based on gradient descent methods similar to those popular for training modern machine learning models, but these can suffer from the existence of local traps that cause the optimisation process to get stuck.
In our new work [https://arxiv.org/abs/2204.08494] we introduce a radical new optimisation method which uses a root finding method to find the the parameters where a large number of covariance functions are all equal to 0, which encodes the solution to our problem, and observe that the method can significantly outperform other methods by orders of magnitude in terms of convergence speed. We measure these functions efficiently using the recent development of the classical shadow method [https://www.nature.com/articles/s41567-020-0932-7]. Using more of these covariances allows us to increase the performance of the optimisation with little extra time on the quantum computer, effectively offloading computation from the quantum computer to a classical processor, a property which is ideal for near term quantum computers which are still limited by noise.
You can find a demo mathematica notebook outlining the use of the method on a 5 qubit spin model here
Figure 1. A slice through the landscape of covariances showing a joint root where all the covariances vanish.
