Chaos and complexity by design
Abstract We study the relationship between quantum chaos and pseudorandomness by developing probes of unitary design. A natural probe of randomness is the “frame poten-tial,” which is minimized by unitary k-designs and measures the 2-norm distance between the Haar random unitary ensemble and another...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2017-04-01
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| Series: | Journal of High Energy Physics |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1007/JHEP04(2017)121 |
| _version_ | 1818280712359378944 |
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| author | Daniel A. Roberts Beni Yoshida |
| author_facet | Daniel A. Roberts Beni Yoshida |
| author_sort | Daniel A. Roberts |
| collection | DOAJ |
| description | Abstract We study the relationship between quantum chaos and pseudorandomness by developing probes of unitary design. A natural probe of randomness is the “frame poten-tial,” which is minimized by unitary k-designs and measures the 2-norm distance between the Haar random unitary ensemble and another ensemble. A natural probe of quantum chaos is out-of-time-order (OTO) four-point correlation functions. We show that the norm squared of a generalization of out-of-time-order 2k-point correlators is proportional to the kth frame potential, providing a quantitative connection between chaos and pseudorandomness. Additionally, we prove that these 2k-point correlators for Pauli operators completely determine the k-fold channel of an ensemble of unitary operators. Finally, we use a counting argument to obtain a lower bound on the quantum circuit complexity in terms of the frame potential. This provides a direct link between chaos, complexity, and randomness. |
| first_indexed | 2024-12-12T23:53:35Z |
| format | Article |
| id | doaj.art-33cdf49d7bff416baaa57f371a3a56dd |
| institution | Directory Open Access Journal |
| issn | 1029-8479 |
| language | English |
| last_indexed | 2024-12-12T23:53:35Z |
| publishDate | 2017-04-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of High Energy Physics |
| spelling | doaj.art-33cdf49d7bff416baaa57f371a3a56dd2022-12-22T00:06:37ZengSpringerOpenJournal of High Energy Physics1029-84792017-04-012017416410.1007/JHEP04(2017)121Chaos and complexity by designDaniel A. Roberts0Beni Yoshida1Center for Theoretical Physics and Department of Physics, Massachusetts Institute of TechnologyPerimeter Institute for Theoretical PhysicsAbstract We study the relationship between quantum chaos and pseudorandomness by developing probes of unitary design. A natural probe of randomness is the “frame poten-tial,” which is minimized by unitary k-designs and measures the 2-norm distance between the Haar random unitary ensemble and another ensemble. A natural probe of quantum chaos is out-of-time-order (OTO) four-point correlation functions. We show that the norm squared of a generalization of out-of-time-order 2k-point correlators is proportional to the kth frame potential, providing a quantitative connection between chaos and pseudorandomness. Additionally, we prove that these 2k-point correlators for Pauli operators completely determine the k-fold channel of an ensemble of unitary operators. Finally, we use a counting argument to obtain a lower bound on the quantum circuit complexity in terms of the frame potential. This provides a direct link between chaos, complexity, and randomness.http://link.springer.com/article/10.1007/JHEP04(2017)121AdS-CFT CorrespondenceGauge-gravity correspondenceRandom SystemsHolography and condensed matter physics (AdS/CMT) |
| spellingShingle | Daniel A. Roberts Beni Yoshida Chaos and complexity by design Journal of High Energy Physics AdS-CFT Correspondence Gauge-gravity correspondence Random Systems Holography and condensed matter physics (AdS/CMT) |
| title | Chaos and complexity by design |
| title_full | Chaos and complexity by design |
| title_fullStr | Chaos and complexity by design |
| title_full_unstemmed | Chaos and complexity by design |
| title_short | Chaos and complexity by design |
| title_sort | chaos and complexity by design |
| topic | AdS-CFT Correspondence Gauge-gravity correspondence Random Systems Holography and condensed matter physics (AdS/CMT) |
| url | http://link.springer.com/article/10.1007/JHEP04(2017)121 |
| work_keys_str_mv | AT danielaroberts chaosandcomplexitybydesign AT beniyoshida chaosandcomplexitybydesign |