  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.
