  Relevant metrological scenarios involve the simultaneous estimation of
multiple parameters. The fundamental ingredient to achieve quantum-enhanced
performances is based on the use of appropriately tailored quantum probes.
However, reaching the ultimate resolution allowed by physical laws requires non
trivial estimation strategies both from a theoretical and a practical point of
view. A crucial tool for this purpose is the application of adaptive learning
techniques. Indeed, adaptive strategies provide a flexible approach to obtain
optimal parameter-independent performances, and optimize convergence to the
fundamental bounds with limited amount of resources. Here, we combine on the
same platform quantum-enhanced multiparameter estimation attaining the
corresponding quantum limit and adaptive techniques. We demonstrate the
simultaneous estimation of three optical phases in a programmable integrated
photonic circuit, in the limited resource regime. The obtained results show the
possibility of successfully combining different fundamental methodologies
towards transition to quantum sensors applications.
