Phase estimation protocols provide a fundamental benchmark for the field of quantum metrology. The latter represents one of the most relevant applications of quantum theory, potentially enabling the capability of measuring unknown physical parameters with improved precision over classical strategies. Within this context, most theoretical and experimental studies have focused on determining the fundamental bounds and how to achieve them in the asymptotic regime where a large number of resources is employed. However, in most applications it is necessary to achieve optimal precisions by performing only a limited number of measurements. To this end, machine learning techniques can be applied as a powerful optimization tool. Here, we implement experimentally single-photon adaptive phase estimation protocols enhanced by machine learning, showing the capability of reaching optimal precision after a small number of trials. In particular, we introduce a new approach for Bayesian estimation that exhibit best performances for very low number of photons N. Furthermore, we study the resilience to noise of the tested methods, showing that the optimized Bayesian approach is very robust in the presence of imperfections. Application of this methodology can be envisaged in the more general multiparameter case, that represents a paradigmatic scenario for several tasks including imaging or Hamiltonian learning.
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