Random unitaries from Hamiltonian dynamics
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911200299712512 |
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| author | Cui, Laura Schuster, Thomas Mao, Liang Huang, Hsin-Yuan Brandao, Fernando |
| author_facet | Cui, Laura Schuster, Thomas Mao, Liang Huang, Hsin-Yuan Brandao, Fernando |
| contents | The nature of randomness and complexity growth in systems governed by unitary dynamics is a fundamental question in quantum many-body physics. This problem has motivated the study of models such as local random circuits and their convergence to Haar-random unitaries in the long-time limit. However, these models do not correspond to any family of physical time-independent Hamiltonians. In this work, we address this gap by studying the indistinguishability of time-independent Hamiltonian dynamics from truly random unitaries. On one hand, we establish a no-go result showing that for any ensemble of constant-local Hamiltonians and any evolution times, the resulting time-evolution unitary can be efficiently distinguished from Haar-random and fails to form a $2$-design or a pseudorandom unitary (PRU). On the other hand, we prove that this limitation can be overcome by increasing the locality slightly: there exist ensembles of random polylog-local Hamiltonians in one-dimension such that under constant evolution time, the resulting time-evolution unitary is indistinguishable from Haar-random, i.e. it forms both a unitary $k$-design and a PRU. Moreover, these Hamiltonians can be efficiently simulated under standard cryptographic assumptions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_08434 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Random unitaries from Hamiltonian dynamics Cui, Laura Schuster, Thomas Mao, Liang Huang, Hsin-Yuan Brandao, Fernando Quantum Physics Statistical Mechanics Strongly Correlated Electrons Computational Complexity Mathematical Physics The nature of randomness and complexity growth in systems governed by unitary dynamics is a fundamental question in quantum many-body physics. This problem has motivated the study of models such as local random circuits and their convergence to Haar-random unitaries in the long-time limit. However, these models do not correspond to any family of physical time-independent Hamiltonians. In this work, we address this gap by studying the indistinguishability of time-independent Hamiltonian dynamics from truly random unitaries. On one hand, we establish a no-go result showing that for any ensemble of constant-local Hamiltonians and any evolution times, the resulting time-evolution unitary can be efficiently distinguished from Haar-random and fails to form a $2$-design or a pseudorandom unitary (PRU). On the other hand, we prove that this limitation can be overcome by increasing the locality slightly: there exist ensembles of random polylog-local Hamiltonians in one-dimension such that under constant evolution time, the resulting time-evolution unitary is indistinguishable from Haar-random, i.e. it forms both a unitary $k$-design and a PRU. Moreover, these Hamiltonians can be efficiently simulated under standard cryptographic assumptions. |
| title | Random unitaries from Hamiltonian dynamics |
| topic | Quantum Physics Statistical Mechanics Strongly Correlated Electrons Computational Complexity Mathematical Physics |
| url | https://arxiv.org/abs/2510.08434 |