Random insights into the complexity of two-dimensional tensor network calculations
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arXiv
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| Autores principales: | , , , , , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866914829721141248 |
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| author | Gonzalez-Garcia, Sofia Sang, Shengqi Hsieh, Timothy H. Boixo, Sergio Vidal, Guifre Potter, Andrew C. Vasseur, Romain |
| author_facet | Gonzalez-Garcia, Sofia Sang, Shengqi Hsieh, Timothy H. Boixo, Sergio Vidal, Guifre Potter, Andrew C. Vasseur, Romain |
| contents | Projected entangled pair states (PEPS) offer memory-efficient representations of some quantum many-body states that obey an entanglement area law, and are the basis for classical simulations of ground states in two-dimensional (2d) condensed matter systems. However, rigorous results show that exactly computing observables from a 2d PEPS state is generically a computationally hard problem. Yet approximation schemes for computing properties of 2d PEPS are regularly used, and empirically seen to succeed, for a large subclass of (not too entangled) condensed matter ground states. Adopting the philosophy of random matrix theory, in this work we analyze the complexity of approximately contracting a 2d random PEPS by exploiting an analytic mapping to an effective replicated statistical mechanics model that permits a controlled analysis at large bond dimension. Through this statistical-mechanics lens, we argue that: i) although approximately sampling wave-function amplitudes of random PEPS faces a computational-complexity phase transition above a critical bond dimension, ii) one can generically efficiently estimate the norm and correlation functions for any finite bond dimension. These results are supported numerically for various bond-dimension regimes. It is an important open question whether the above results for random PEPS apply more generally also to PEPS representing physically relevant ground states |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_11053 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Random insights into the complexity of two-dimensional tensor network calculations Gonzalez-Garcia, Sofia Sang, Shengqi Hsieh, Timothy H. Boixo, Sergio Vidal, Guifre Potter, Andrew C. Vasseur, Romain Quantum Physics Other Condensed Matter Statistical Mechanics Projected entangled pair states (PEPS) offer memory-efficient representations of some quantum many-body states that obey an entanglement area law, and are the basis for classical simulations of ground states in two-dimensional (2d) condensed matter systems. However, rigorous results show that exactly computing observables from a 2d PEPS state is generically a computationally hard problem. Yet approximation schemes for computing properties of 2d PEPS are regularly used, and empirically seen to succeed, for a large subclass of (not too entangled) condensed matter ground states. Adopting the philosophy of random matrix theory, in this work we analyze the complexity of approximately contracting a 2d random PEPS by exploiting an analytic mapping to an effective replicated statistical mechanics model that permits a controlled analysis at large bond dimension. Through this statistical-mechanics lens, we argue that: i) although approximately sampling wave-function amplitudes of random PEPS faces a computational-complexity phase transition above a critical bond dimension, ii) one can generically efficiently estimate the norm and correlation functions for any finite bond dimension. These results are supported numerically for various bond-dimension regimes. It is an important open question whether the above results for random PEPS apply more generally also to PEPS representing physically relevant ground states |
| title | Random insights into the complexity of two-dimensional tensor network calculations |
| topic | Quantum Physics Other Condensed Matter Statistical Mechanics |
| url | https://arxiv.org/abs/2307.11053 |