Quantum Simulation with a Boson Sampling Circuit

In this work we study a system that consists of $2M$ matter qubits that interact through a boson sampling circuit, i.e., an $M$-port interferometer, embedded in two different architectures. We prove that, under the conditions required to derive a master equation, the qubits evolve according to effective bipartite XY spin Hamiltonians, with or without local and collective dissipation terms. This opens the door to the simulation of any bipartite spin or hard-core boson models and exploring dissipative phase transitions as the competition between coherent and incoherent exchange of excitations. We also show that in the purely dissipative regime this model has a large number of exact and approximate dark states, whose structure and decay rates can be estimated analytically. We finally argue that this system may be used for the adiabatic preparation of boson sampling states encoded in the matter qubits.