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arXiv:2511.08543

Design boosters: from constant-time quantum chaos to $\infty$-designs and beyond

Soumik Ghosh, Arjun Mirani, Yihui Quek, Michelle Xu

quant-phcond-mat.stat-mechhep-thSubmitted Nov 11, 2025• Announced Nov 12, 2025• 0 citations
v1 · Nov 11, 2025v2 · Oct 5, 2026
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Abstract

We study a counterintuitive property of 'conditioning' on the result of measuring a subsystem of a quantum state: such conditioning can boost design quality, at the cost of increased system size. We work in the setting of deep thermalization from many-body physics: starting from a bipartite state on a global system $(A,B)$ drawn from a $k$-design, we measure system $B$ in the computational basis, keep the outcome and examine the state that remains in system $A$, approximating the overall ensemble (the 'projected ensemble') by a $k'$-design. We ask: how does the design quality change due to this procedure, or how does $k'$ compare to $k$? We give the first rigorous example of unitary dynamics generating a state such that, projection at very early (constant) times can boost design randomness. These dynamics are those of quantum chaos, modeled by the evolution of a Hamiltonian drawn from the Gaussian Unitary Ensemble (GUE). We show that, even though a state generated by such dynamics at constant time only forms a $k=\mathcal{O}(1)$ design, the projected ensemble is Haar-random (or a $k'=\infty$ design) in the thermodynamic limit (i.e. when $N_B=\infty$). This phenomenon persists even with weaker and more physically realistic assumptions; our results can be appropriately applied to non-GUE Hamiltonians that nevertheless show likely chaotic signatures in their eigenbases. Moreover, we show that with no assumption on how the global state was generated, a $k$-design experiences a degradation in design quality to $k' = \lfloor k/2 \rfloor$. This improves upon best prior results on the deep thermalization of designs. Together, our contributions argue for design boosting as a result of chaos and showcase a novel mechanism to generate good designs.

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41 pages, 2 figures

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Design boosters: from constant-time quantum chaos to ∞\infty∞-designs and beyond
We study a counterintuitive property of 'conditioning' on the result of measuring a subsystem of a quantum state: such conditioning can boost design quality, at the cost of increased system size. We work in the setting of deep thermalization from many-body physics: starting from a bipartite state on a global system (A,B)(A,B)(A,B) drawn from a kkk-design, we measure system BBB in the computational basis, keep the outcome and examine the state that remains in system AAA, approximating the overall ensemble (the 'projected ensemble') by a k′k'k′-design. We ask: how does the design quality change due to this procedure, or how does k′k'k′ compare to kkk? We give the first rigorous example of unitary dynamics generating a state such that, projection at very early (constant) times can boost design randomness. These dynamics are those of quantum chaos, modeled by the evolution of a Hamiltonian drawn from the Gaussian Unitary Ensemble (GUE). We show that, even though a state generated by such dynamics at constant time only forms a k=O(1)k=\mathcal{O}(1)k=O(1) design, the projected ensemble is Haar-random (or a k′=∞k'=\inftyk′=∞ design) in the thermodynamic limit (i.e. when NB=∞N_B=\inftyNB​=∞). This phenomenon persists even with weaker and more physically realistic assumptions; our results can be appropriately applied to non-GUE Hamiltonians that nevertheless show likely chaotic signatures in their eigenbases. Moreover, we show that with no assumption on how the global state was generated, a kkk-design experiences a degradation in design quality to k′=⌊k/2⌋k' = \lfloor k/2 \rfloork′=⌊k/2⌋. This improves upon best prior results on the deep thermalization of designs. Together, our contributions argue for design boosting as a result of chaos and showcase a novel mechanism to generate good designs.
Abstract The projected ensemble is based on the study of the quantum state of a subsystem A conditioned on projective measurements in its complement. Recent studies have observed that a more refined measure of the thermalization of a chaotic quantum system can be defined on the basis of convergence of the projected ensemble to a quantum state design, i.e. a system thermalizes when it becomes indistinguishable, up to the k th moment, from a Haar ensemble of uniformly distributed pure states. Here we consider a random unitary circuit with the brick-wall geometry and analyze its convergence to the Haar ensemble through the frame potential and its mapping to a statistical mechanical problem. This approach allows us to highlight a geometric interpretation of the frame potential based on the existence of a fluctuating membrane, similar to those appearing in the study of entanglement entropies. At large local Hilbert space dimension q , we find that all moments converge simultaneously with a time scaling linearly in the size of region A , a feature previously observed in dual unitary models. However, based on the geometric interpretation, we argue that the scaling at finite q on the basis of rare membrane fluctuations, finding the logarithmic scaling of design times t k = O ( log ⁡ k ) . Our results are supported with numerical simulations performed at q = 2.
We study the emergence over time of a universal, uniform distribution of quantum states supported on a finite subsystem, induced by projectively measuring the rest of the system. Dubbed deep thermalization , this phenomenon represents a form of equilibration in quantum many-body systems stronger than regular thermalization, which only constrains the ensemble-averaged values of observables. While there exist quantum circuit models of dynamics in one dimension where this phenomenon can be shown to arise exactly, these are special in that deep thermalization occurs at precisely the same time as regular thermalization. Here, we present an exactly-solvable model of chaotic dynamics where the two processes can be shown to occur over different time scales. The model is composed of a finite subsystem coupled to an infinite random-matrix bath through a small constriction, and highlights the role of locality and imperfect thermalization in constraining the formation of such universal wavefunction distributions. We test our analytical predictions against exact numerical simulations, finding excellent agreement.
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.
Quantum thermalization in a many-body system is defined by the approach of local subsystems toward a universal form, describable as an ensemble of quantum states wherein observables acquire thermal expectation values.Recently, it has been demonstrated that the distribution of these quantum states can also exhibit universal statistics, upon associating each state with the outcome of a local projective measurement of the complementary subsystem.Specifically, this collection of pure quantum states-called the projected ensemble-can, under certain conditions, mimic the behavior of a maximally entropic uniformly random ensemble, i.e., form a quantum state design, representing a "deeper" form of quantum thermalization.In this work, we investigate the dynamical process underlying this novel emergent universality.Leveraging a space-time duality mapping for one-dimensional quantum circuits, we argue that the physics of dynamical purification, which arises in the context of monitored quantum systems, constrains the approach of the projected ensemble toward the uniform distribution.We prove that the absence of dynamical purification in the space-time dual dynamics (a condition realized in dual-unitary quantum circuits with appropriate initial states and final measurement bases) generically yields exact state designs for all moments k at the same time, extending previous rigorous results [Ho and Choi, Phys.Rev. Lett.128, 060601 (2022)].Conversely, we show that, departing from these conditions, dynamical purification can lead to a separation of time scales between the formation of a quantum state design for moment k = 1 (regular thermalization) and for high moments k 1 (deep thermalization).Our results suggest that the projected ensemble can probe nuanced features of quantum dynamics inaccessible to regular thermalization, such as quantum information scrambling.
Quantum many-body systems provide a unique platform for exploring the rich interplay between chaos, randomness, and complexity. In a recently proposed paradigm known as deep thermalization, random quantum states of system A are generated by performing projective measurements on system B following chaotic Hamiltonian evolution acting jointly on AB. In this scheme, the randomness of the projected state ensemble arises from the intrinsic randomness of the outcomes when B is measured. Here, we propose a modified scheme in which classical randomness injected during the protocol is converted by quantum chaos into quantum randomness of the resulting state ensemble. We show that for generic chaotic systems this conversion is optimal in that each bit of injected classical entropy generates as much additional quantum randomness as adding an extra qubit to B. This significantly enhances the randomness of the projected ensemble without imposing additional demands on the quantum hardware. Our scheme can be easily implemented on typical analog quantum simulators, providing a more scalable route for generating quantum randomness valuable for many applications. In particular, we demonstrate that the accuracy of a shadow tomography protocol can be substantially improved.
Measuring part of a many-body wave function generates a projected ensemble of pure quantum states on the unmeasured subsystem. Recent advances in deep thermalization have shown that, in chaotic systems, this ensemble universally converges to the maximally random ensemble compatible with its average density matrix, known as the Scrooge ensemble. By contrast, free-fermion systems are nonchaotic, and their quantum states are constrained to remain Gaussian. Motivated by this distinction, we introduce the fermionic Gaussian Scrooge ensemble to describe deep thermalization in generic free-fermion systems. This ensemble is defined as a distortion of the Gaussian Haar ensemble by the average density matrix, and its moments reveal an enlarged symmetry of Gaussian states in the replicated Hilbert space. We demonstrate the emergence of the fermionic Gaussian Scrooge ensemble in two complementary settings: (1) we analytically prove that the projected ensemble of the SYK2_22​ model follows the fermionic Gaussian Scrooge ensemble at arbitrary evolution times; (2) we provide numerical evidence that it emerges at sufficiently long times in random Gaussian circuits with charge conservation. Our results establish the fermionic Gaussian Scrooge ensemble as a universal description of deep thermalization in free-fermion systems.