Randomization Times under Quantum Chaotic Hamiltonian Evolution

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.

Figures (4)

• Figure 1: (a) Schematic of the theoretical setup. We sample initially unentangled states and evolve them under a generic quantum-chaotic Hamiltonian $H$. At each time $t$, we measure the full distribution of a chosen subsystem observable $X$ (e.g., $X:$ Renyi entropy, entanglement entropy, or subsystem energy). (b) Example: distribution of entanglement entropy $S_A$ of time-evolved states relative to the average entropy of Haar-random states $\tilde{S}_A$ ($\delta S_A = S_A - \tilde{S}_A$). Shown are the mean $\bar{S}_A$ over $M=100$ initial states (solid lines) along with the associated standard deviation $\sigma_{S_A}$ across samples (shaded region surrounding $\bar{S}_A$). For comparison, we indicate with a shaded region the size of Haar fluctuations $\tilde{\sigma}_{S_A}$. At times larger than $\tau_L \approx 20/\varepsilon_*$, the evolved states reproduce both the mean and statistical fluctuations characteristic of Haar-random states ($\varepsilon_*$ denotes the characteristic local energy scale of $H$).
• Figure 2: Quantum state randomization as measured through entanglement observables. (a) Relaxation of half-system entanglement observables to their Haar expectation values, shown for system sizes $L = 12, 14, \ldots, 20$ in the mixed-field Ising model (MFIM). We plot the deviation $\delta X(t) = \bar{X}(t) - \tilde{X}$, rescaled by the typical fluctuations $\tilde{\sigma}_X$. Data is shown for the entanglement entropy ($X=S_A$, solid lines) and second Renyi entropy ($X=S_2$, dash-dotted lines), with $\bar{X}$ averaged over 100 realizations, and with $\tilde{X}$ and $\tilde{\sigma}_X$ corresponding to the average and fluctuations of $X$ for Haar-random states. The shaded region serves as a guide to the eye to indicate the timescale where $\delta X = \tilde{\sigma}_X$. Parameters: $g = 1.05$ and $h = 0.4$. (b) Randomization time $\tau_L$, defined as the timescale at which $\delta X(\tau_L) = \sigma_X$, obtained from $X=S_A$ and $X=S_2$ for subsystem sizes $L_A = 2,4,\ldots,L/2$ (contiguous sites). The bars indicate the variability of $\tau_L$ for across $X$ and $L_A$ values. For comparison, we show data of $\tau_L$ in random quantum circuits (RQCs): $\tau_L \sim O(L)$ for circuits without a conservation law and $\tau_L \sim O(L^2)$ for U(1)-conserving RQCs.
• Figure 3: Dynamics of the full counting statistics of energy within a subsystem of size $L_A = 4$ embedded in a chain of $L = 16$ spins. (a) Shown is the distribution of $p_{E_A}$ for three representative energy levels $E_A$ and $M=100$ initial conditions, illustrating exponential-in-time convergence to their Haar-like values (dotted-dashed lines) on the same timescale $\tau_L$ identified in Fig. \ref{['fig:finitesizescaling']}. The shaded regions indicate the variance $\sigma_{p_{E_A}}$ of $p_{E_A}$ across different initializations. The inset shows the distribution $p_{E_A}$ for the 16 energy levels in subsystem $A$ at time $t=0$ and in the long-time limit $t \to \infty$, for a single initialization. (b) System-size scaling of the timescales at which the fluctuations of $p_{E_A}$ reach their late-time value, averaged over all energy outcomes $E_A$ in $A$.
• Figure 4: Randomization of time evolved states, as measured through the relaxation of the entanglement entropy distribution, as a function of the transverse field $g$ in the MFIM. Results are shown for $M = 10^2$ samples of the half-system entanglement entropy in a chain of size $L = 16$, with $g$ varied in increments of $\Delta g = 0.05$. The black curve corresponds to the parameter value with the fastest randomization, $g_* = 1.00\pm 0.05$, while the progressively lighter shades denote increasing $|g - g_*|$. The shaded region indicates the typical entanglement fluctuations expected for Haar-random states. The inset displays the randomization time as a function of $g$, normalized by system size.