Randomization Times under Quantum Chaotic Hamiltonian Evolution

TL;DR

This paper investigates how generic quantum-chaotic Hamiltonian evolution can generate effective randomness rapidly, with local and nonlocal observables reaching Haar-random expectations on polynomial timescales, surpassing previous limitations.

Contribution

It demonstrates that non-random, physically realistic Hamiltonians can produce Haar-like randomness efficiently, challenging assumptions about the necessity of random circuits for rapid thermalization.

Findings

Effective Haar-randomization occurs before ergodic exploration of Hilbert space.

Local and nonlocal observables equilibrate to Haar expectations quickly.

Randomization timescales are linear in system size, bypassing sub-ballistic entropy growth.

Abstract

Randomness generation through quantum-chaotic evolution underpins foundational questions in statistical mechanics and applications across quantum information science, including benchmarking, tomography, metrology, and demonstrations of quantum computational advantage. While statistical mechanics successfully captures the temporal averages of local observables, understanding randomness at the level of higher statistical moments remains a daunting challenge, with analytic progress largely confined to random quantum circuit models or fine-tuned systems exhibiting space-time duality. Here we study how much randomness can be dynamically generated by generic quantum-chaotic evolution under physical, non-random Hamiltonians. Combining theoretical insights with numerical simulations, we show that for broad classes of initially unentangled states, the dynamics become effectively Haar-random well before the system can ergodically explore the physically accessible Hilbert space. Both local and highly nonlocal observables, including entanglement measures, equilibrate to their Haar expectation values and fluctuations on polynomial timescales with remarkably high numerical precision, and with the fastest randomization occurring in regions of parameter space previously identified as maximally chaotic. Interestingly, this effective randomization can occur on timescales linear in system size, suggesting that the sub-ballistic growth of Renyi entropies typically observed in systems with conservation laws can be bypassed in non-random Hamiltonians with an appropriate choice of initial conditions.