Quantum complexity of time evolution with chaotic Hamiltonians
Abstract We study the quantum complexity of time evolution in large-N chaotic systems, with the SYK model as our main example. This complexity is expected to increase linearly for exponential time prior to saturating at its maximum value, and is related to the length of minimal geodesics on the mani...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2020-01-01
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| Series: | Journal of High Energy Physics |
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| Online Access: | https://doi.org/10.1007/JHEP01(2020)134 |
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| author | Vijay Balasubramanian Matthew DeCross Arjun Kar Onkar Parrikar |
| author_facet | Vijay Balasubramanian Matthew DeCross Arjun Kar Onkar Parrikar |
| author_sort | Vijay Balasubramanian |
| collection | DOAJ |
| description | Abstract We study the quantum complexity of time evolution in large-N chaotic systems, with the SYK model as our main example. This complexity is expected to increase linearly for exponential time prior to saturating at its maximum value, and is related to the length of minimal geodesics on the manifold of unitary operators that act on Hilbert space. Using the Euler-Arnold formalism, we demonstrate that there is always a geodesic between the identity and the time evolution operator e−iHt whose length grows linearly with time. This geodesic is minimal until there is an obstruction to its minimality, after which it can fail to be a minimum either locally or globally. We identify a criterion — the Eigenstate Complexity Hypothesis (ECH) — which bounds the overlap between off- diagonal energy eigenstate projectors and the k-local operators of the theory, and use it to argue that the linear geodesic will at least be a local minimum for exponential time. We show numerically that the large-N SYK model (which is chaotic) satisfies ECH and thus has no local obstructions to linear growth of complexity for exponential time, as expected from holographic duality. In contrast, we also study the case with N = 2 fermions (which is integrable) and find short-time linear complexity growth followed by oscillations. Our analysis relates complexity to familiar properties of physical theories like their spectra and the structure of energy eigenstates and has implications for the hypothesized computational complexity class separations PSPACE BQP/poly and PSPACE BQSUBEXP/subexp, and the “fast-forwarding” of quantum Hamiltonians. |
| format | Article |
| id | doaj-art-a11593ee180a4e8a9570a25f0327f6bf |
| institution | Kabale University |
| issn | 1029-8479 |
| language | English |
| publishDate | 2020-01-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of High Energy Physics |
| spelling | doaj-art-a11593ee180a4e8a9570a25f0327f6bf2025-02-09T12:06:42ZengSpringerOpenJournal of High Energy Physics1029-84792020-01-012020114410.1007/JHEP01(2020)134Quantum complexity of time evolution with chaotic HamiltoniansVijay Balasubramanian0Matthew DeCross1Arjun Kar2Onkar Parrikar3David Rittenhouse Laboratory, University of PennsylvaniaDavid Rittenhouse Laboratory, University of PennsylvaniaDavid Rittenhouse Laboratory, University of PennsylvaniaDavid Rittenhouse Laboratory, University of PennsylvaniaAbstract We study the quantum complexity of time evolution in large-N chaotic systems, with the SYK model as our main example. This complexity is expected to increase linearly for exponential time prior to saturating at its maximum value, and is related to the length of minimal geodesics on the manifold of unitary operators that act on Hilbert space. Using the Euler-Arnold formalism, we demonstrate that there is always a geodesic between the identity and the time evolution operator e−iHt whose length grows linearly with time. This geodesic is minimal until there is an obstruction to its minimality, after which it can fail to be a minimum either locally or globally. We identify a criterion — the Eigenstate Complexity Hypothesis (ECH) — which bounds the overlap between off- diagonal energy eigenstate projectors and the k-local operators of the theory, and use it to argue that the linear geodesic will at least be a local minimum for exponential time. We show numerically that the large-N SYK model (which is chaotic) satisfies ECH and thus has no local obstructions to linear growth of complexity for exponential time, as expected from holographic duality. In contrast, we also study the case with N = 2 fermions (which is integrable) and find short-time linear complexity growth followed by oscillations. Our analysis relates complexity to familiar properties of physical theories like their spectra and the structure of energy eigenstates and has implications for the hypothesized computational complexity class separations PSPACE BQP/poly and PSPACE BQSUBEXP/subexp, and the “fast-forwarding” of quantum Hamiltonians.https://doi.org/10.1007/JHEP01(2020)134AdS-CFT CorrespondenceField Theories in Lower Dimensions |
| spellingShingle | Vijay Balasubramanian Matthew DeCross Arjun Kar Onkar Parrikar Quantum complexity of time evolution with chaotic Hamiltonians Journal of High Energy Physics AdS-CFT Correspondence Field Theories in Lower Dimensions |
| title | Quantum complexity of time evolution with chaotic Hamiltonians |
| title_full | Quantum complexity of time evolution with chaotic Hamiltonians |
| title_fullStr | Quantum complexity of time evolution with chaotic Hamiltonians |
| title_full_unstemmed | Quantum complexity of time evolution with chaotic Hamiltonians |
| title_short | Quantum complexity of time evolution with chaotic Hamiltonians |
| title_sort | quantum complexity of time evolution with chaotic hamiltonians |
| topic | AdS-CFT Correspondence Field Theories in Lower Dimensions |
| url | https://doi.org/10.1007/JHEP01(2020)134 |
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