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We analyze the energy and training data requirements for supervised learningof an $M$-mode linear optical circuit by minimizing an empirical risk definedsolely from the action of the circuit on coherent states. When the linearoptical circuit acts non-trivially only on $k<M$ unknown modes (i.e., a linearoptical $k$-junta), we provide an energy-efficient, adaptive algorithm thatidentifies the junta set and learns the circuit. We compare two schemes forallocating 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 secondscheme, a single random $MT$-mode coherent state with energy $E$ is partitionedinto $T$ training coherent states. The latter scheme exhibits a polynomialadvantage in training data size sufficient for convergence of the empiricalrisk to the full risk due to concentration of measure on the $(2MT-1)$-sphere.Specifically, generalization bounds for both schemes are proven, which indicatethe sufficiency of $O(E^{1/2}M)$ training states ($O(E^{1/3}M^{1/3})$ trainingstates) in the first (second) scheme.

Learning linear optical circuits with coherent states