Heisenberg scaling precision in the estimation of functions of parameters in linear optical networks

We propose a metrological strategy reaching Heisenberg-scaling precision in the estimation of functions of any fixed number p of arbitrary parameters encoded in a generic M-channel linear network. This scheme is experimentally feasible since it only employs a single-mode squeezed vacuum and homodyne detection on a single output channel. Two auxiliary linear networks are required and their role is twofold: to refocus the signal into a single channel after the interaction with the interferometer, and to fix the function of the parameters to be estimated according to the linear network analyzed. Although the refocusing requires some knowledge on the parameters, we show that the required precision on the prior measurement is achievable with a classic measurement. We conclude by discussing two paradigmatic schemes in which the choice of the auxiliary stages allows us to change the function of the unknown parameter to estimate.