We study computational problems related to the Schr\"odinger operator $H = -\Delta + V$ 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 Schr\"odinger operator implements universal quantum computation, i.e., it is BQP-hard, and (ii) estimating the ground energy of the Schr\"odinger 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 $\texttt{StoqMA}\varsubsetneq \texttt{QMA}$.

翻译：我们研究了实空间中薛定谔算子$H = -\Delta + V$相关的计算问题，其中满足以下条件：(i) 势函数$V$是光滑的，其值及其导数在$n$的某个多项式范围内有界；(ii) $V$仅包含$O(1)$体相互作用。我们证明了：(i) 模拟由薛定谔算子生成的动力学可实现通用量子计算，即该问题是BQP难的；(ii) 估计薛定谔算子的基态能量与估计无符号问题（即随机哈密顿量）的局域哈密顿量基态能量具有相同难度，即该问题是StoqMA完全的。这一结果尤为引人关注，因为已知一般玻色子哈密顿量的基态能量问题是QMA难的，且学界普遍认为$\texttt{StoqMA}\varsubsetneq \texttt{QMA}$。