Certificates of quantum many-body properties assisted by machine learning

Computationally intractable tasks are often encountered in physics and optimization. They usually comprise a cost function to be optimized over a so-called feasible set, which is specified by a set of constraints. This may yield, in general, to difficult and nonconvex optimization tasks. A number of standard methods are used to tackle such problems: variational approaches focus on parametrizing a subclass of solutions within the feasible set. In contrast, relaxation techniques have been proposed to approximate it from outside, thus complementing the variational approach to provide ultimate bounds to the global optimal solution. In this paper, we propose a novel approach combining the power of relaxation techniques with deep reinforcement learning in order to find the best possible bounds within a limited computational budget. We illustrate the viability of the method in two paradigmatic problems in quantum physics and quantum information processing: finding the ground state energy of many-body quantum systems, and building energy-based entanglement witnesses of quantum local Hamiltonians. We benchmark our approach against other classical optimization algorithms such as breadth-first search or Monte Carlo, and we characterize the effect of transfer learning. We find the latter may be indicative of phase transitions with a completely autonomous approach. Finally, we provide tools to tackle other common applications in the field of quantum information processing with our method.