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.
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.
The emergence of randomness from unitary quantum dynamics is a central problem across diverse disciplines, ranging from the foundations of statistical mechanics to quantum algorithms and quantum computation. Physical systems are invariably subject to constraints -- from simple scalar symmetries to more complex non-Abelian ones -- that restrict the accessible regions of Hilbert space and obstruct the generation of pure random states. In this work, we show that for systems with noncommuting symmetries such as SU(2), the degree of Haar-like randomization achievable under unitary dynamics is strongly constrained by experimental limitations on state initialization, in particular low-entanglement initial states, rather than by the symmetry-constrained dynamics themselves. Specifically, we show that time-evolved states can, in principle, reproduce Haar-like behavior at the level of finite statistical moments (i.e., those accessible under realistic experimental conditions with a finite number of state copies) provided that the initial state matches the corresponding moments of the conserved operators in the Haar ensemble. However, for the unentangled initial states commonly used in programmable quantum systems, this condition cannot be satisfied. Consequently, even at asymptotically long times in strongly quantum-chaotic regimes, late-time states remain distinguishable from Haar-random states in probes such as entanglement entropy, with deviations from Haar behavior that remain finite with increasing system size. We quantify the maximal entanglement entropy achievable and identify the unentangled initial conditions that yield the most entropic late-time states. Our results show that the combination of non-Abelian symmetry structure and experimental constraints on state preparation can strongly limit the degree of Haar-like randomization achievable at late times.
Initially unentangled states undergoing quantum chaotic dynamics are expected to evolve into featureless states at late times. While this expectation holds true on an averaged, coarse-grained level, it is unclear if this expectation applies to higher statistical moments, as symmetries present in physical systems constrain the exploration of Hilbert space. Here, we study the universal structure of late-time ensembles obtained from unitary dynamics in quantum chaotic systems with symmetries, such as charge or energy conservation. We find that while quantum states do not ergodically explore the entire Hilbert space at late times, the late-time ensemble typically becomes indistinguishable from Haar random states in the thermodynamic limit and at the level of finite statistical moments. Importantly, our results apply to initial states that are easy to prepare in ongoing experiments---specifically, product states---that lie in the middle of the spectrum of quantum chaotic systems. We show that typically these states not only exhibit the same late-time ensemble average as Haar random states, but also the same state-to-state fluctuations and higher statistical moments. In other words, there is no measurement---whether local or nonlocal---that one can perform at the level of finite statistical moments to tell that states are not exploring the entire Hilbert space. Interestingly, within the class of low-entanglement initial states, we also find atypical initial conditions in the middle of the spectrum of Hamiltonians known to be ``maximally chaotic''. Such atypical initial states have smaller variance of the symmetry operator relative to that of Haar random states, and evolve into nonuniversal ensembles that can be distinguished from the Haar ensemble by simple measurements or subsystem properties. In the limiting case of initial states with negligible variance of the symmetry operator (e.g., states with fixed particle number or energy eigenstates), the late-time ensemble has universal behavior captured by constrained random state ensembles. On the one hand, our results reveal that an extremely high level of quantum state randomness can typically be achieved even when dynamics are constrained by symmetries; on the other hand, our results show that one can still achieve strongly nonergodic dynamics at ``infinite temperature'' within quantum chaotic regimes.
Despite its long history, a canonical formulation of quantum ergodicity that applies to general classes of quantum dynamics, including driven systems, has not been fully established. Here we introduce and study a notion of quantum ergodicity for closed systems with time-dependent Hamiltonians, defined as statistical randomness exhibited in their longtime dynamics. Concretely, we consider the temporal ensemble of quantum states (time-evolution operators) generated by the evolution, and investigate the conditions necessary for them to be statistically indistinguishable from uniformly random states (operators) in the Hilbert space (space of unitaries). We find that the number of driving frequencies underlying the Hamiltonian needs to be sufficiently large for this to occur. Conversely, we show that statistical —indistinguishability up to some large but finite moment—can already be achieved by a quantum system driven with a single frequency, i.e., a Floquet system, as long as the driving period is sufficiently long. Our work relates the complexity of a time-dependent Hamiltonian and that of the resulting quantum dynamics, and offers a fresh perspective to the established topics of quantum ergodicity and chaos from the lens of quantum information. Published by the American Physical Society 2024
Information scrambling, the process by which quantum information spreads and becomes effectively inaccessible, is central to modern quantum statistical physics and quantum chaos. These lecture notes provide an introduction to information scrambling from both static and dynamical perspectives. The spectral properties of reduced density matrices arising from Haar-random states are developed through the geometry of the unitary group and the universal results of random matrix theory. This geometric framework yields universal, model-independent predictions for entanglement and spectral statistics, capturing generic features of quantum chaos without reference to microscopic details. Dynamical diagnostics such as the spectral form factor and out-of-time-ordered correlators further reveal the onset of chaos in time-dependent evolution.
The notes are aimed at advanced undergraduate and graduate students in physics, mathematics, and computer science who are interested in the connections between quantum chaos, information dynamics, and quantum computing. Concepts such as entanglement growth, Haar randomness, random-matrix statistics, and unitary t-designs are introduced through their realization in random quantum circuits and circuit complexity. The same mathematical framework also underpins modern quantum-device benchmarking, where approximate unitary designs and random circuits translate geometric ideas into quantitative tools for assessing fidelity, noise, and scrambling efficiency in real quantum processors.
Chaotic quantum many-body dynamics typically lead to relaxation of local observables. In this process, known as quantum thermalization, a subregion reaches a thermal state due to quantum correlations with the remainder of the system, which acts as an intrinsic bath. While the bath is generally assumed to be unobserved, modern quantum science experiments have the ability to track both subsystem and bath at a microscopic level. Here, by utilizing this ability, we discover that measurement results associated with small subsystems exhibit universal random statistics following chaotic quantum many-body dynamics, a phenomenon beyond the standard paradigm of quantum thermalization. We explain these observations with an ensemble of pure states, defined via correlations with the bath, that dynamically acquires a close to random distribution. Such random ensembles play an important role in quantum information science, associated with quantum supremacy tests and device verification, but typically require highly-engineered, time-dependent control for their preparation. In contrast, our approach uncovers random ensembles naturally emerging from evolution with a time-independent Hamiltonian. As an application of this emergent randomness, we develop a benchmarking protocol which estimates the many-body fidelity during generic chaotic evolution and demonstrate it using our Rydberg quantum simulator. Our work has wide ranging implications for the understanding of quantum many-body chaos and thermalization in terms of emergent randomness and at the same time paves the way for applications of this concept in a much wider context.
Abstract Scrambling is a process by which the state of a quantum system is effectively randomized due to the global entanglement that “hides” initially localized quantum information. Closely related notions include quantum chaos and thermalization. Such phenomena play key roles in the study of quantum gravity, many-body physics, quantum statistical mechanics, quantum information etc. Scrambling can exhibit different complexities depending on the degree of randomness it produces. For example, notice that the complete randomization implies scrambling, but the converse does not hold; in fact, there is a significant complexity gap between them. In this work, we lay the mathematical foundations of studying randomness complexities beyond scrambling by entanglement properties. We do so by analyzing the generalized (in particular Rényi) entanglement entropies of designs, i.e. ensembles of unitary channels or pure states that mimic the uniformly random distribution (given by the Haar measure) up to certain moments. A main collective conclusion is that the Rényi entanglement entropies averaged over designs of the same order are almost maximal. This links the orders of entropy and design, and therefore suggests Rényi entanglement entropies as diagnostics of the randomness complexity of corresponding designs. Such complexities form a hierarchy between information scrambling and Haar randomness. As a strong separation result, we prove the existence of (state) 2-designs such that the Rényi entanglement entropies of higher orders can be bounded away from the maximum. However, we also show that the min entanglement entropy is maximized by designs of order only logarithmic in the dimension of the system. In other words, logarithmic-designs already achieve the complexity of Haar in terms of entanglement, which we also call max-scrambling. This result leads to a generalization of the fast scrambling conjecture, that max-scrambling can be achieved by physical dynamics in time roughly linear in the number of degrees of freedom. This paper is an extended version of Phys. Rev. Lett. 120 (2018) 130502 [1].