Bounding the finite-size error of quantum many-body dynamics simulations
Finite-size errors (FSEs), the discrepancies between an observable in a finite system and in the thermodynamic limit, are ubiquitous in numerical simulations of quantum many-body systems. Although a rough estimate of these errors can be obtained from a sequence of finite-size results, a strict, quan...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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American Physical Society
2021-08-01
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| Series: | Physical Review Research |
| Online Access: | http://doi.org/10.1103/PhysRevResearch.3.L032047 |
| _version_ | 1797210937566953472 |
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| author | Zhiyuan Wang Michael Foss-Feig Kaden R. A. Hazzard |
| author_facet | Zhiyuan Wang Michael Foss-Feig Kaden R. A. Hazzard |
| author_sort | Zhiyuan Wang |
| collection | DOAJ |
| description | Finite-size errors (FSEs), the discrepancies between an observable in a finite system and in the thermodynamic limit, are ubiquitous in numerical simulations of quantum many-body systems. Although a rough estimate of these errors can be obtained from a sequence of finite-size results, a strict, quantitative bound on the magnitude of FSE is still missing. Here we derive rigorous upper bounds on the FSE of local observables in real-time quantum dynamics simulations initialized from a product state. In d-dimensional locally interacting systems with a finite local Hilbert space, our bound implies |〈S[over ̂](t)〉_{L}−〈S[over ̂](t)〉_{∞}|≤C(2vt/L)^{cL−μ}, with v, C, c, μ constants independent of L and t, which we compute explicitly. For periodic boundary conditions (PBCs), the constant c is twice as large as that for open boundary conditions (OBCs), suggesting that PBCs have smaller FSEs than OBCs at early times. The bound can be generalized to a large class of correlated initial states as well. As a byproduct, we prove that the FSE of local observables in ground-state simulations decays exponentially with L under a suitable spectral gap condition. Our bounds are practically useful in determining the validity of finite-size results, as we demonstrate in simulations of the one-dimensional (1D) quantum Ising and Fermi-Hubbard models. |
| first_indexed | 2024-04-24T10:18:32Z |
| format | Article |
| id | doaj.art-81553e5e94084844a2b4c1e5c2575d9a |
| institution | Directory Open Access Journal |
| issn | 2643-1564 |
| language | English |
| last_indexed | 2024-04-24T10:18:32Z |
| publishDate | 2021-08-01 |
| publisher | American Physical Society |
| record_format | Article |
| series | Physical Review Research |
| spelling | doaj.art-81553e5e94084844a2b4c1e5c2575d9a2024-04-12T17:13:01ZengAmerican Physical SocietyPhysical Review Research2643-15642021-08-0133L03204710.1103/PhysRevResearch.3.L032047Bounding the finite-size error of quantum many-body dynamics simulationsZhiyuan WangMichael Foss-FeigKaden R. A. HazzardFinite-size errors (FSEs), the discrepancies between an observable in a finite system and in the thermodynamic limit, are ubiquitous in numerical simulations of quantum many-body systems. Although a rough estimate of these errors can be obtained from a sequence of finite-size results, a strict, quantitative bound on the magnitude of FSE is still missing. Here we derive rigorous upper bounds on the FSE of local observables in real-time quantum dynamics simulations initialized from a product state. In d-dimensional locally interacting systems with a finite local Hilbert space, our bound implies |〈S[over ̂](t)〉_{L}−〈S[over ̂](t)〉_{∞}|≤C(2vt/L)^{cL−μ}, with v, C, c, μ constants independent of L and t, which we compute explicitly. For periodic boundary conditions (PBCs), the constant c is twice as large as that for open boundary conditions (OBCs), suggesting that PBCs have smaller FSEs than OBCs at early times. The bound can be generalized to a large class of correlated initial states as well. As a byproduct, we prove that the FSE of local observables in ground-state simulations decays exponentially with L under a suitable spectral gap condition. Our bounds are practically useful in determining the validity of finite-size results, as we demonstrate in simulations of the one-dimensional (1D) quantum Ising and Fermi-Hubbard models.http://doi.org/10.1103/PhysRevResearch.3.L032047 |
| spellingShingle | Zhiyuan Wang Michael Foss-Feig Kaden R. A. Hazzard Bounding the finite-size error of quantum many-body dynamics simulations Physical Review Research |
| title | Bounding the finite-size error of quantum many-body dynamics simulations |
| title_full | Bounding the finite-size error of quantum many-body dynamics simulations |
| title_fullStr | Bounding the finite-size error of quantum many-body dynamics simulations |
| title_full_unstemmed | Bounding the finite-size error of quantum many-body dynamics simulations |
| title_short | Bounding the finite-size error of quantum many-body dynamics simulations |
| title_sort | bounding the finite size error of quantum many body dynamics simulations |
| url | http://doi.org/10.1103/PhysRevResearch.3.L032047 |
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