On the Computational Complexity of Schrödinger Operators

We study computational problems related to the Schrödinger operator $H=-\mathrm{\Delta }+V$ in the real space under the condition that (i) the potential function $V$ is smooth and has its value and derivative bounded within some polynomial of $n$ and (ii) $V$ only consists of $O(1)$-body interactions. We prove that (i) simulating the dynamics generated by the Schrödinger operator implements universal quantum computation, i.e., it is BQP-hard, and (ii) estimating the ground energy of the Schrödinger operator is as hard as estimating that of local Hamiltonians with no sign problem (a.k.a. stoquastic Hamiltonians), i.e., it is StoqMA-complete. This result is particularly intriguing because the ground energy problem for general bosonic Hamiltonians is known to be QMA-hard and it is widely believed that $\mathtt{\text{StoqMA}}⊊\mathtt{\text{QMA}}$.