Hilbert-space ergodicity (HSE) provides a new framework for studying thermalization in driven quantum systems, complementing the eigenstate thermalization hypothesis, which is restricted to static systems. This ergodicity is hierarchical: by quantifying how randomly the dynamics explores the Hilbert space, one obtains a family of levels termed $k$-HSE. While HSE has been observed at the lowest and highest levels, finite-order HSE dynamics remains largely unexplored due to the difficulty of constructing such drives. Here, we explore this intermediate regime and uncover its distinctive physics. We first propose and prove that a family of $m$-tone drives on qubits realizes $k$-HSE up to $k = 2m{-}3$, with drive parameters determined at $O(k)$ cost. Using a single nitrogen-vacancy center in diamond, we verify this design by showing that a 3-tone drive realizes 3-HSE, with fourth-order statistics depending on the initial state. Further in-depth theoretical analysis shows that this initial-state dependence is generic across drives, demonstrating the possibility of recovering the initial state from higher-order statistics even when the dynamics is ergodic. Our work broadens the study of quantum ergodicity and reveals intriguing physics within its hierarchy.
Unitary $k$-designs provide a resource-efficient framework for emulating Haar randomness up to the $k$-th order. Quench-based protocols have recently been shown to generate such designs, but achieving this typically requires multiple Hamiltonian realizations, even when using temporal ensembles. Here, we show that no independent Hamiltonians are required: a single chaotic Hamiltonian is sufficient to generate approximate unitary $k$-designs, even when that Hamiltonian is spatially local. We introduce a two-Pauli-kick (2PK) protocol, in which unitary evolution under a fixed Hamiltonian is interspersed with two Pauli operator insertions (kicks). We find that the resulting frame potential approaches the Haar value at long times, with deviations suppressed by the inverse Hilbert-space dimension. We verify this for single realizations of Gaussian random matrices, the Majorana and Spin Sachdev-Ye-Kitaev models, and deterministic local quantum spin chains. Remarkably, in deterministic spin systems, the 2PK protocol generates approximate unitary designs in regimes where quench-based protocols are either inapplicable or fail to converge. Furthermore, our protocol provides a finite-temperature extension of the frame potential and establishes an analytic bound in terms of the equilibrium partition function. We discuss a holographic perspective on this mechanism.
Deep thermalization concerns the emergence of universal pure-state statistics in projected ensembles at late times, yet the mechanism governing the initial growth of randomness remains unclear. Here, we study the short-time dynamics of projected ensembles generated from initially unentangled states, quantifying their randomness using frame potentials. For arbitrary Hamiltonians, provided the initial bath state has full support in the measurement basis, we show that the frame potentials to cubic order in time are determined entirely by the subsystem purity, and hence by the bipartite entanglement generated between the unmeasured subsystem and its complement. The entanglement timescale therefore sets the initial timescale for the growth of local randomness, independently of the bath measurement basis. For unitarily invariant Hamiltonian ensembles, we further relate the projected-ensemble frame potential at order $k$ to the $4k$-point spectral form factor, or equivalently to the $2k$th frame potential of the global unitary dynamics, establishing a direct connection between local and global randomness. Our results identify entanglement as the mechanism governing the onset of randomness in projected ensembles and clarify how local randomness emerges from global quantum dynamics.
We challenge the widespread consensus that mean-field theory fails to describe long-range open quantum systems in the presence of symmetry breaking and/or when starting from strongly correlated states (e.g., macroscopic superpositions). While recent literature relies on cumulant expansions to capture such systems, this approach rests on truncations with no clear justification. Here, we show that it is, at best, conceptually redundant in the strong long-range regime. We show that the evolution can be decomposed into, and fully reconstructed from, independent mean-field dynamics. This decomposition generates the entire hierarchy of cumulants and, as a byproduct, identifies---to our knowledge, for the first time---a regime in which cumulant expansions exactly predict low-order cumulants. We illustrate the power of our findings with two applications: we compute the moment generating function for nonequilibrium $\mathcal{Z}_2$ symmetry breaking, and construct states restoring time-translation symmetry in time crystals. In both cases, our method fully reproduces the exact many-body dynamics, which is out of reach of cumulant expansions. Our results reclaim the exactness of mean-field theory, offering a transparent framework for large-scale open quantum systems.
Quantum complexity has emerged as a natural probe of chaos, thermalization, and the black hole interior on timescales long after local observables have equilibrated. However, its time evolution has been studied almost exclusively in subsystems confined to a single connected region. We show, using complementary tools from holography and random quantum circuits, that noncontiguous subsystems composed of multiple disjoint regions give rise to qualitatively new physics compared to the contiguous case. First, at finite temperature, a subsystem occupying less than half of the total system can carry high complexity at late times, even as the complexity of its larger complement remains low. This inversion of the usual hierarchy is intrinsically thermal: it vanishes in the infinite-temperature limit, which is the regime modeled by random quantum circuits. Second, if a subsystem's complexity equilibrates at an early time, then fragmenting it into $m$ disjoint components can further reduce this timescale by a factor of $m$, a phenomenon we exhibit in both holography and random quantum circuits. These results not only sharpen the correspondence between geometric and computational notions of complexity, but also motivate the search for novel complexity phenomena in quantum dynamics.
Yale Fan, Nicholas Hunter-Jones, A. Karch et al.· 0 citations
Monitored many-body quantum systems can undergo sharp learnability transitions characterized by how much information can be learned by the observer. When the dynamics conserves a non-Abelian charge, such as an $SU(2)$ spin, understanding how the observer learns the total charge remains an outstanding problem. Unlike the Abelian case, where charge measurements on distinct sites commute, the $SU(2)$-symmetric readouts are noncommuting fusion measurements, making learning a genuinely quantum inference problem. In this work, we propose a theory of $1+1d$ monitored quantum dynamics with $SU(2)$ symmetry, and show that it can be described by an effective replicated loop model comprised of a replica-pairing field and a diffusive ($z=2$) background sector that carries the $SU(2)$ charge and remains gapless throughout the phase diagram. Our theory predicts that the"spin-sharpening''and entanglement transitions coincide as a single transition. Ordering of the pairing field produces volume-law entanglement and hides the background sector from measurements, leading to a learning time of $t\sim L^{3}$ for the total spin. When the pairing field disorders, the background sector alone gives logarithmic entanglement and a diffusive learning time $t\sim L^{2}$. Our analysis is controlled by a large-loop-fugacity expansion.
Programmable quantum devices nominally act on a Hilbert space whose dimension grows exponentially with the number of constituents, but the presence of noise makes it unlikely that they remain coherent across all of this immense Hilbert space. Then, what is the effective coherent quantum dimension that should be associated with such imperfect devices? To answer this question we here introduce an operational basis-independent framework which imposes a dimension bottleneck on the programmable transformations. Concretely we ask how strongly the quantum information they process can be compressed. Formalizing this idea we identify three inequivalent notions, termed $d$-compressibility, $d$-simulability and $d$-embeddability, which differ in the causal structure used to impose the bottleneck and form a strict hierarchy. The framework unifies several existing notions: joint measurability and simulability of quantum measurements, and the absolute dimensionality of state ensembles, are recovered as special cases. We illustrate the hierarchy with noisy qubit measurements in complementary bases, and we determine the white-noise thresholds at which the set of all noisy unitary channels in dimension $n$, a noisy universal quantum processor, becomes $d$-compressible, $d$-simulable and $d$-embeddable. The thresholds confirm the expectation -- maintaining coherence across the full Hilbert space becomes increasingly demanding as the nominal dimension $n$ increases.
Pavel Sekatski· 0 citations
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