Unitary designs provide efficient substitutes for Haar-random unitaries in quantum information processing, randomized measurements, and many-body quantum dynamics. We propose a protocol based on time evolutions of a single chaotic Hamiltonian interleaved with unitary perturbations to generate unitary designs. We derive the frame potential of the resulting ensemble in terms of those of the intermediate ensemble. The intermediate ensemble need not itself be Haar random or exhibit Haar-level frame potentials, provided that its trace moments and frame-potential growth are sufficiently suppressed. The nontrivial Pauli set and the Clifford ensemble are simple examples, whereas ensembles whose cardinality does not grow with the Hilbert-space dimension generally fail. We further establish a frame-potential correspondence between multi-Hamiltonian temporal protocols and one-Hamiltonian protocols interleaved with repeated fixed trace-suppressed perturbations, showing that two perturbations suffice to recover Haar-level frame potentials. For local Hamiltonians, we derive a lower bound on the space--depth control resource, while more generally we establish a universal finite-window bound that constrains the required temporal sampling range. Together, these results identify complementary spatial and temporal resource requirements for perturbation-assisted randomness generation.
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.
Unitary
k-designs are central to quantum information and quantum many-body physics, as they provide efficient proxies for Haar-random dynamics. We study how chaotic Hamiltonian evolution can generate unitary
k-designs using the frame potential (FP). Unlike standard approaches based on independently sampled Hamiltonians or fine-tuned evolution times, we consider a quenched temporal ensemble in which the Hamiltonians are sampled once and then held fixed, while randomness enters only through the evolution times. We compare a two-step protocol (2SP), with evolution under two fixed Hamiltonians, to a three-step protocol (3SP) with one additional quench, with three evolution times sampled independently. Analytically and numerically, we show that the 2SP fails to realize general unitary
k-designs, whereas the 3SP realizes them for arbitrary
k. The random phases in the 3SP impose stronger index-matching constraints in the FP, eliminating the independent permutation degrees of freedom that still remain in the 2SP. Even with imperfect time averaging, the 3SP reaches a given accuracy within a parametrically shorter time window. These results hold universally across different symmetry classes and persist in both all-to-all and local models with random interactions.
We present a single-quench protocol that generates unitary
k-designs with minimal control. A system first evolves under a random Hamiltonian
H1; at a switch time
ts≥tTh (the Thouless time), it is quenched to an independently drawn
H2 from the same ensemble and then evolves under
H2. This single quench breaks residual spectral correlations that prevent strictly time-independent chaotic dynamics from forming higher-order designs. The resulting ensemble approaches a unitary
k-design using only a single control operation -- far simpler than Brownian schemes with continuously randomized couplings or protocols that apply random quenches at short time intervals. Beyond offering a direct route to Haar-like randomness, the protocol yields an operational, measurement-friendly definition of
tTh and provides a quantitative diagnostic of chaoticity. It further enables symmetry-resolved and open-system extensions, circuit-level single-quench analogs, and immediate applications to randomized measurements, benchmarking, and tomography.
We study the generation of state
k-designs from time evolution under a fixed Hamiltonian. Specifically, we consider the ensemble
E={e−iHt∣ψ0⟩∣ t∼Unif[0,T],∣ψ0⟩∼E′}, where the initial states are sampled from an ensemble
E′. For Hamiltonians drawn from the Gaussian unitary ensemble, we derive a simple relation between the frame potential of the evolved ensemble
E and that of the initial ensemble
E′ in the large evolution time limit. This relation shows that
E forms an exact state
k-design in the thermodynamic limit as long as
E′ forms a state 1-design. Remarkably, we further show, both analytically and numerically, that time evolution under a simple nonintegrable mixed-field Ising Hamiltonian can generate approximate state
k-designs with high precision, starting from product states in an appropriately chosen Pauli basis. We also analyze the finite-
T correction and find it scales as
O(1/T). To reduce the evolution time, we propose an
M-step quench protocol that suppresses this correction to
O(1/TM), which is also verified numerically. We then extend our analysis to unitary ensembles, deriving an analogous recursion relation for the unitary frame potential. Our results elucidate the mechanisms underlying recent proposals for generating unitary
k-designs through sequential quantum quenches in a unified manner.
We present a single-quench protocol that generates unitary k-designs with minimal control. A system first evolves under a random Hamiltonian H_{1}; at a switch time t_{s}≥t_{Th} (the Thouless time), it is quenched to an independently drawn H_{2} from the same ensemble and then evolves under H_{2}. This single quench breaks residual spectral correlations that prevent strictly time-independent chaotic dynamics from forming higher-order designs. The resulting ensemble approaches a unitary k-design using only a single control operation-far simpler than Brownian schemes with continuously randomized couplings or protocols that apply random quenches at short time intervals. Our analytical results are derived for Gaussian unitary random matrices, while our numerical demonstrations are carried out primarily in the Sachdev-Ye-Kitaev model. Beyond offering a direct route to Haar-like randomness, the protocol yields an operational, measurement-friendly definition of t_{Th} and provides a quantitative diagnostic of chaoticity. It further suggests symmetry-resolved and open-system extensions, circuit-level single-quench analogs, and applications to randomized measurements, benchmarking, and tomography.