Random unitaries from Hamiltonian dynamics

• URL: http://arxiv.org/abs/2510.08434v1
• Date: Thu, 09 Oct 2025 16:46:55 GMT
• Title: Random unitaries from Hamiltonian dynamics
• Authors: Laura Cui, Thomas Schuster, Liang Mao, Hsin-Yuan Huang, Fernando Brandao,
• Abstract summary: We study the indistinguishability of time-independent Hamiltonian dynamics from truly random unitaries.<n>We prove that 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.
• Score: 37.08488662212577
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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