Learning linear optical circuits with coherent states

We analyze the energy and training data requirements for supervised learning of an M -mode linear optical circuit by minimizing an empirical risk defined solely from the action of the circuit on coherent states. When the linear optical circuit acts non-trivially only on k < M unknown modes (i.e. a linear optical k -junta), we provide an energy-efficient, adaptive algorithm that identifies the junta set and learns the circuit. We compare two schemes for allocating a total energy, E , to the learning algorithm. In the first scheme, each of the T random training coherent states has energy E / T . In the second scheme, a single random MT -mode coherent state with energy E is partitioned into T training coherent states. The latter scheme exhibits a polynomial advantage in training data size sufficient for convergence of the empirical risk to the full risk due to concentration of measure on the ( 2 M T − 1 ) -sphere. Specifically, generalization bounds for both schemes are proven, which indicate that for ε -approximation of the full risk by the empirical risk with high probability, O ( E 2 / 3 M 2 / 3 / ϵ 2 / 3 ) training states are sufficient for the first scheme and O ( E 1 / 3 M 1 / 3 / ϵ 2 / 3 ) training states are sufficient for the second scheme.