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.Introduction. -Quantum metrology is one of the most promising applications of quantum theory [1][2][3][4][5], where the aim is to obtain enhanced performances in the estimation of unknown physical parameters by employing quantum resources. A notable benchmark for quantum metrology is provided by phase estimation, a task where the parameter to be measured is an optical phase embedded within an interferometric setup. In this scenario, an input probe field is prepared in a suitable state and sent through the system. The value of the phase is retrieved by measuring the field after the evolution in the interferometer, and by repeating the procedure N times to perform statistical analysis. While the ultimate precision achievable with classical resources is known to be bounded by the standard quantum limit (SQL), stating that the achievable error on the unknown phase φ scales as N −1/2 (being N the number of photons), the adoption of quantum inputs can in principle improve the performances up to the Heisenberg limit (HL) [1,2], scaling as N −1 . Several theoretical and experimental studies [6][7][8][9][10][11][12][13][14] focused on devising experimental schemes able to reach quantum enhanced performances. Furthermore, recent advances in integrated photonics has opened new possibilities for the implementation and the development of phase estimation protocols [15][16][17][18][19][20][21][22]. In parallel, a thorough investigation has been dedicated to identifying the effect of experimental noise and losses [23][24][25][26]. In the scenario where the parameter to be estimated is a single phase, it is always possible to identify the op...

Phase estimation represents a significant example to test the application of quantum theory for enhanced measurements of unknown physical parameters. Several recipes have been developed, allowing to define strategies to reach the ultimate bounds in the asymptotic limit of a large number of trials. However, in certain applications it is crucial to reach such bound when only a small number of probes is employed. Here, we discuss an asymptotically optimal, machine learning based, adaptive single-photon phase estimation protocol that allows us to reach the standard quantum limit when a very limited number of photons is employed.

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