On the Computational Complexity of Schrödinger Operators

• URL: http://arxiv.org/abs/2411.05120v1
• Date: Thu, 07 Nov 2024 19:39:52 GMT
• Title: On the Computational Complexity of Schrödinger Operators
• Authors: Yufan Zheng, Jiaqi Leng, Yizhou Liu, Xiaodi Wu,
• Abstract summary: We study computational problems related to the Schr"odinger operator $H = -Delta + V$ in the real space. 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) This result is particularly intriguing because the ground energy problem for general bosonic Hamiltonians is known
• Score: 6.1436827446807705
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• Abstract: 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 $\texttt{StoqMA}\varsubsetneq \texttt{QMA}$.

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