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| Format: | Preprint |
| Veröffentlicht: |
2023
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| Online-Zugang: | https://arxiv.org/abs/2302.00173 |
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| _version_ | 1866913358701133824 |
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| author | Pan, Ruizhi Clark, Charles W. |
| author_facet | Pan, Ruizhi Clark, Charles W. |
| contents | Neural-network state representations of quantum many-body systems are attracting great attention and more rigorous quantitative analysis about their expressibility and complexity is warranted. Our analysis of the restricted Boltzmann machine (RBM) state representation of one-dimensional (1D) quantum spin systems provides new insight into their computational complexity. We define a class of long-range-fast-decay (LRFD) RBM states with quantifiable upper bounds on truncation errors and provide numerical evidence for a large class of 1D quantum systems that may be approximated by LRFD RBMs of at most polynomial complexities. These results lead us to conjecture that the ground states of a wide range of quantum systems may be exactly represented by LRFD RBMs or a variant of them, even in cases where other state representations become less efficient. At last, we provide the relations between multiple typical state manifolds. Our work proposes a paradigm for doing complexity analysis for generic long-range RBMs which naturally yields a further classification of this manifold. This paradigm and our characterization of their nonlocal structures may pave the way for understanding the natural measure of complexity for quantum many-body states described by RBMs and are generalizable for higher-dimensional systems and deep neural-network quantum states. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_00173 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Efficiency of neural-network state representations of one-dimensional quantum spin systems Pan, Ruizhi Clark, Charles W. Quantum Physics Disordered Systems and Neural Networks Neural-network state representations of quantum many-body systems are attracting great attention and more rigorous quantitative analysis about their expressibility and complexity is warranted. Our analysis of the restricted Boltzmann machine (RBM) state representation of one-dimensional (1D) quantum spin systems provides new insight into their computational complexity. We define a class of long-range-fast-decay (LRFD) RBM states with quantifiable upper bounds on truncation errors and provide numerical evidence for a large class of 1D quantum systems that may be approximated by LRFD RBMs of at most polynomial complexities. These results lead us to conjecture that the ground states of a wide range of quantum systems may be exactly represented by LRFD RBMs or a variant of them, even in cases where other state representations become less efficient. At last, we provide the relations between multiple typical state manifolds. Our work proposes a paradigm for doing complexity analysis for generic long-range RBMs which naturally yields a further classification of this manifold. This paradigm and our characterization of their nonlocal structures may pave the way for understanding the natural measure of complexity for quantum many-body states described by RBMs and are generalizable for higher-dimensional systems and deep neural-network quantum states. |
| title | Efficiency of neural-network state representations of one-dimensional quantum spin systems |
| topic | Quantum Physics Disordered Systems and Neural Networks |
| url | https://arxiv.org/abs/2302.00173 |