Non-unitary free boson dynamics and the boson sampling problem

Abstract: We explore the free boson unitary dynamics subject to repeated random forced measurement. The input state is chosen as a Fock state in real space with the particle number conserved in the entire dynamics. We show that each boson is performing a non-unitary quantum walk in real space and its dynamics can be mapped to directed polymers in a random medium with complex amplitude. We numerically show that in the one dimensional system, when the measurement strength is finite, the system is in the frozen phase with all the bosons localized in the real space. Furthermore, these single particle wave functions take the same probability distribution in real space after long time evolution. Due to this property, the boson sampling for the output state becomes easy to solve. We further investigate circuit with non-local unitary dynamics and numerically demonstrate that there could exist a phase transition from a localized phase to a delocalized phase by varying the measurement strength.