On the Computational Complexity of Schrödinger Operators
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866913574193987584 |
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| author | Zheng, Yufan Leng, Jiaqi Liu, Yizhou Wu, Xiaodi |
| author_facet | Zheng, Yufan Leng, Jiaqi Liu, Yizhou Wu, Xiaodi |
| contents | We study computational problems related to the Schrödinger operator $H = -Δ+ V$ 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 $\texttt{StoqMA}\varsubsetneq \texttt{QMA}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_05120 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Computational Complexity of Schrödinger Operators Zheng, Yufan Leng, Jiaqi Liu, Yizhou Wu, Xiaodi Quantum Physics Computational Complexity Mathematical Physics We study computational problems related to the Schrödinger operator $H = -Δ+ V$ 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 $\texttt{StoqMA}\varsubsetneq \texttt{QMA}$. |
| title | On the Computational Complexity of Schrödinger Operators |
| topic | Quantum Physics Computational Complexity Mathematical Physics |
| url | https://arxiv.org/abs/2411.05120 |