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.
Quantum simulators with disordered spin Hamiltonians offer a natural route to Haar-like unitary ensembles, but face a competition. Disorder enhances randomness, yet strong disorder suppresses dynamics and localizes the system in the computational basis. This suggests an optimal intermediate disorder strength for approximating a unitary
k-design, which has so far been identified mainly empirically. Here, we develop a perturbative theory for the moment gap
Δ(k), the deviation of the
k-th moment operator from its Haar value, for the full spectrum of the interacting XXZ chain in a fixed-magnetization sector. We find the non-monotonic asymptotic scalings
lnΔ(k)∼−σ2 at weak disorder and
lnΔ(k)∼−1/σ at strong disorder, whose crossover sets an optimal disorder strength
σc∼Jxy, where
Jxy is the energy scale of the non-random interaction. We further show that a multi-quench protocol further suppresses the gap exponentially in the number of quenches. Numerical simulations confirm both scalings and this exponential suppression, and we demonstrate the usefulness of the protocol for classical shadow tomography. This work suggests that strong disorder may block both thermalization and the formation of unitary designs through a common localization 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.
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.
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.
The realization of unitary designs is of fundamental interest in quantum science and typically requires the ability to implement structured quantum circuits. Recent developments have explored the possibility of generating unitary designs using only a small number of quantum quenches, in which the evolution during each interval is governed by a static Hamiltonian. In particular, it has been established that at least three chaotic Hamiltonians are required when only Hamiltonian evolutions are employed. In this work, we propose the emergence of unitary designs in the temporal ensemble of qubit systems evolved under two distinct chaotic Hamiltonians for sufficiently long times, supplemented by an intermediate random Pauli operation inserted between them. This result follows from the universal Pauli spectrum of chaotic Hamiltonians, a central concept in the study of non-stabilizerness. Our theoretical predictions are verified numerically using explicit examples, including Gaussian unitary ensemble Hamiltonians and random spin models. We further investigate finite-time and finite-size corrections to the protocol. Our results provide new insights into the dynamical generation of quantum randomness and offer a new route toward realizing unitary designs in chaotic systems.