A minimal set of measurement operators for quantum state tomography has in the non-degenerate case ideally eigenbases which are mutually unbiased. This is different for the degenerate case. Here, we consider the situation where the measurement operators are projections on individual pure quantum states. This corresponds to maximal degeneracy. We present numerically optimized sets of projectors and find that they significantly outperform those which are taken from a set of mutually unbiased bases.

Finding optimal measurement schemes in quantum state tomography is a fundamental problem in quantum computation. It is known that for non-degenerate operators the optimal measurement scheme is based on mutually unbiassed bases. This paper is a follow up from our previous work, where we use standard numerical approaches to look for optimal measurement schemes, where the measurement operators are projections on individual pure quantum states. In this paper we demonstrate the usefulness of several machine learning techniques – reinforcement learning and parallel machine learning approaches, to discover measurement schemes, which are significantly better than the ones discovered by standard numerical methods in our previous work. The high-performing quorums of projection operators we have discovered have complex structure and symmetries, which may imply that the optimal solution will possess such symmetries.