Learning linear optical circuits with coherent states

Abstract

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 (2MT−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(E2/3M2/3/ϵ2/3) training states are sufficient for the first scheme and O(E1/3M1/3/ϵ2/3) training states are sufficient for the second scheme.