Complexity of sampling as an order parameter

We consider the classical complexity of approximately simulating time evolution under spatially local quadratic bosonic Hamiltonians for time t. We obtain upper bounds on the scaling of t with the number of bosons, n, for which simulation is classically efficient. We also obtain a lower bound on the scaling of t with n for which this problem reduces to a general instance of the boson sampling problem and is hence hard, assuming the conjectures of Aaronson and Arkhipov [Proc. 43rd Annu. ACM Symp. Theory Comput. STOC '11]. We view these results in the light of classifying phases of physical systems based on parameters in the Hamiltonian and conjecture a link to dynamical phase transitions. In doing so, we combine ideas from mathematical physics and computational complexity to gain insight into the behavior of condensed matter systems.