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 are employed. However, in most applications, it is necessary to achieve optimal precision 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 an approach for Bayesian estimation that exhibits best performance for a 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, which represents a paradigmatic scenario for several tasks, including imaging or Hamiltonian learning.

• Received 22 December 2017
• Revised 27 June 2018

DOI:https://doi.org/10.1103/PhysRevApplied.10.044033