Shedding light on classical shadows: learning photonic quantum states

Abstract: Efficient learning of quantum state properties is both a fundamental and practical problem in quantum information theory. Classical shadows have emerged as an efficient method for estimating properties of unknown quantum states, with rigorous statistical guarantees, by performing randomized measurement on a few number of copies. With the advent of photonic technologies, formulating efficient learning algorithms for such platforms comes out as a natural problem. Here, we introduce a classical shadow protocol for learning photonic quantum states via randomized passive linear optical transformations and photon-number measurement. We show that this scheme is efficient for a large class of observables of interest. We experimentally demonstrate our findings on a twelve-mode photonic integrated quantum processing unit. Our protocol allows for scalable learning of a wide range of photonic state properties and paves the way to applying the already rich variety of applications of classical shadows to photonic platforms.