Certain properties and applications of shallow bosonic circuits

We introduce a novel approach to solve optimization problems on a boson sampling device assisted by classical machine-learning techniques. By virtue of the parity function, we map all measurement patterns, which label the basis spanning an $M$-mode bosonic Hilbert space, to the Hilbert space of $M$ qubits. As a result, the sampled probability function can be interpreted as a result of sampling a multiqubit circuit. The method is presented on several instances of a QUBO/Ising problem as well as portfolio optimization problems. Among many demonstrated properties of the parity function is the ability to chart the entire qubit Hilbert space no matter how shallow the initial bosonic circuits is. In order to show this we link boson sampling circuits to a class of finite Young's lattices (a special poset with the so-called Ferrers diagrams ordered by inclusion), Boolean lattices and the properties of Dyck/staircase paths on integer lattices. Our results and methods can be applied to a large variety of photonic circuits, including the deep ones of essentially any geometry, but our main focus is on shallow circuits as they are less affected by photon loss and relatively easy to implement in the form of a time-bin interferometer.