Function estimation protocols with photonic sensors

A method for operating a quantum sensor network (200) to estimate a linear function of a set of local physical parameters (204), the method comprising:

• identifying a set of integer vectors (234) W, wherein each vector ω in the set of integer vectors (234) defines a photon number distribution satisfying a norm constraint relative to a fixed total photon number N distributed across a plurality of optical modes (202);
• calculating a protocol vector (236) r comprising non-negative integers that satisfy a linear Diophantine equation relating a matrix W formed from the set of integer vectors (234) to a vector of rational coefficients α representing the linear function;
• generating a photonic probe state (212) initialized in a superposition of a first branch and a second branch according to a first vector selected from the set of integer vectors (234);
• encoding the set of local physical parameters (204) into the photonic probe state (212) by subjecting the photonic probe state (212) to a sequence of passes through the plurality of optical modes (202);
• applying a control unitary operation (218) to the photonic probe state (212) between the passes to coherently switch the photonic probe state (212) from the first vector to a subsequent vector selected from the set of integer vectors (234) based on the protocol vector (236), wherein the control unitary operation (218) maintains the relative phase between the first branch and the second branch;
• and detecting the relative phase accumulated on the photonic probe state (212) after a total number of passes M defined by the protocol vector (236).