Linear growth of circuit complexity from Brownian dynamics
Abstract How rapidly can a many-body quantum system generate randomness? Using path integral methods, we demonstrate that Brownian quantum systems have circuit complexity that grows linearly with time. In particular, we study Brownian clusters of N spins or fermions with time-dependent all-to-all in...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2023-08-01
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| Series: | Journal of High Energy Physics |
| Subjects: | |
| Online Access: | https://doi.org/10.1007/JHEP08(2023)190 |
| _version_ | 1797647444369997824 |
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| author | Shao-Kai Jian Gregory Bentsen Brian Swingle |
| author_facet | Shao-Kai Jian Gregory Bentsen Brian Swingle |
| author_sort | Shao-Kai Jian |
| collection | DOAJ |
| description | Abstract How rapidly can a many-body quantum system generate randomness? Using path integral methods, we demonstrate that Brownian quantum systems have circuit complexity that grows linearly with time. In particular, we study Brownian clusters of N spins or fermions with time-dependent all-to-all interactions, and calculate the Frame Potential to characterize complexity growth in these models. In both cases the problem can be mapped to an effective statistical mechanics problem which we study using path integral methods. Within this framework it is straightforward to show that the kth Frame Potential comes within ϵ of the Haar value after a time of order t ~ kN + k log k + log ϵ −1. Using a bound on the diamond norm, this implies that such circuits are capable of coming very close to a unitary k-design after a time of order t ~ kN. We also consider the same question for systems with a time-independent Hamiltonian and argue that a small amount of time-dependent randomness is sufficient to generate a k-design in linear time provided the underlying Hamiltonian is quantum chaotic. These models provide explicit examples of linear complexity growth that are analytically tractable and are directly applicable to practical applications calling for unitary k-designs. |
| first_indexed | 2024-03-11T15:16:26Z |
| format | Article |
| id | doaj.art-09b31a0705ae415fbe5fee1596013010 |
| institution | Directory Open Access Journal |
| issn | 1029-8479 |
| language | English |
| last_indexed | 2024-03-11T15:16:26Z |
| publishDate | 2023-08-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of High Energy Physics |
| spelling | doaj.art-09b31a0705ae415fbe5fee15960130102023-10-29T12:11:19ZengSpringerOpenJournal of High Energy Physics1029-84792023-08-012023814210.1007/JHEP08(2023)190Linear growth of circuit complexity from Brownian dynamicsShao-Kai Jian0Gregory Bentsen1Brian Swingle2Department of Physics and Engineering Physics, Tulane UniversityDepartment of Physics, Brandeis UniversityDepartment of Physics, Brandeis UniversityAbstract How rapidly can a many-body quantum system generate randomness? Using path integral methods, we demonstrate that Brownian quantum systems have circuit complexity that grows linearly with time. In particular, we study Brownian clusters of N spins or fermions with time-dependent all-to-all interactions, and calculate the Frame Potential to characterize complexity growth in these models. In both cases the problem can be mapped to an effective statistical mechanics problem which we study using path integral methods. Within this framework it is straightforward to show that the kth Frame Potential comes within ϵ of the Haar value after a time of order t ~ kN + k log k + log ϵ −1. Using a bound on the diamond norm, this implies that such circuits are capable of coming very close to a unitary k-design after a time of order t ~ kN. We also consider the same question for systems with a time-independent Hamiltonian and argue that a small amount of time-dependent randomness is sufficient to generate a k-design in linear time provided the underlying Hamiltonian is quantum chaotic. These models provide explicit examples of linear complexity growth that are analytically tractable and are directly applicable to practical applications calling for unitary k-designs.https://doi.org/10.1007/JHEP08(2023)190Matrix ModelsNon-Equilibrium Field TheoryRandom Systems |
| spellingShingle | Shao-Kai Jian Gregory Bentsen Brian Swingle Linear growth of circuit complexity from Brownian dynamics Journal of High Energy Physics Matrix Models Non-Equilibrium Field Theory Random Systems |
| title | Linear growth of circuit complexity from Brownian dynamics |
| title_full | Linear growth of circuit complexity from Brownian dynamics |
| title_fullStr | Linear growth of circuit complexity from Brownian dynamics |
| title_full_unstemmed | Linear growth of circuit complexity from Brownian dynamics |
| title_short | Linear growth of circuit complexity from Brownian dynamics |
| title_sort | linear growth of circuit complexity from brownian dynamics |
| topic | Matrix Models Non-Equilibrium Field Theory Random Systems |
| url | https://doi.org/10.1007/JHEP08(2023)190 |
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