Quantum complexity and generalized area law in fully connected models
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910836439646208 |
|---|---|
| author | Kim, Donghoon Kuwahara, Tomotaka |
| author_facet | Kim, Donghoon Kuwahara, Tomotaka |
| contents | The area law for entanglement entropy fundamentally reflects the complexity of quantum many-body systems, demonstrating ground states of local Hamiltonians to be represented with low computational complexity. While this principle is well-established in one-dimensional systems, little is known beyond 1D cases, and attempts to generalize the area law on infinite-dimensional graphs have largely been disproven. In this work, for non-critical ground states of Hamiltonians on fully connected graphs, we establish a generalized area law up to a polylogarithmic factor in system size, by effectively reducing the boundary area to a constant scale for interactions between subsystems. This result implies an efficient approximation of the ground state by the matrix product state up to an approximation error of $1/\text{poly}(n)$. As the core technique, we develop the mean-field renormalization group approach, which rigorously guarantees efficiency by systematically grouping regions of the system and iteratively approximating each as a product state. This approach provides a rigorous pathway to efficiently simulate ground states of complex systems, advancing our understanding of infinite-dimensional quantum many-body systems and their entanglement structures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_02140 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantum complexity and generalized area law in fully connected models Kim, Donghoon Kuwahara, Tomotaka Quantum Physics Quantum Gases Statistical Mechanics Mathematical Physics The area law for entanglement entropy fundamentally reflects the complexity of quantum many-body systems, demonstrating ground states of local Hamiltonians to be represented with low computational complexity. While this principle is well-established in one-dimensional systems, little is known beyond 1D cases, and attempts to generalize the area law on infinite-dimensional graphs have largely been disproven. In this work, for non-critical ground states of Hamiltonians on fully connected graphs, we establish a generalized area law up to a polylogarithmic factor in system size, by effectively reducing the boundary area to a constant scale for interactions between subsystems. This result implies an efficient approximation of the ground state by the matrix product state up to an approximation error of $1/\text{poly}(n)$. As the core technique, we develop the mean-field renormalization group approach, which rigorously guarantees efficiency by systematically grouping regions of the system and iteratively approximating each as a product state. This approach provides a rigorous pathway to efficiently simulate ground states of complex systems, advancing our understanding of infinite-dimensional quantum many-body systems and their entanglement structures. |
| title | Quantum complexity and generalized area law in fully connected models |
| topic | Quantum Physics Quantum Gases Statistical Mechanics Mathematical Physics |
| url | https://arxiv.org/abs/2411.02140 |