Inference-based quantum sensing

In a standard Quantum Sensing (QS) task one aims at estimating an unknown parameter encoded into an n-qubit probe state, via measurements of the system. The success of this task hinges on the ability to correlate changes in the parameter to changes in the system response (i.e., changes in the measurement outcomes). For simple cases the form of the system response is known, but the same cannot be said for realistic scenarios, as no general closed-form expression exists. In this work we present an inference-based scheme for QS. We show that, for a general class of unitary families of encoding the response function can be fully characterized by only performing measurements at 2n+1 parameters. This allows us to infer the value of an unknown parameter given the measured response, as well as to determine the sensitivity of the scheme, which characterizes its overall performance. We show that the inference error can be well controlled if one measures the system response using a number of shots that scales poly-logarithmically with the system size. Furthermore, the framework presented can be broadly applied as it remains valid for arbitrary probe states and measurement schemes, and, even holds in the presence of quantum noise. We also discuss how to extend our results beyond unitary families. Finally, to showcase our method we implement it for a QS task on real quantum hardware, and in numerical simulations.