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arXiv:2610.18954

Optimal Disorder Strength for Approximate Unitary $k$-Designs in Spin Chains

Tian-Gang Zhou, Yi-Neng Zhou, Thierry Giamarchi, Julian Sonner

quant-phcond-mat.dis-nncond-mat.quant-gascond-mat.stat-mechSubmitted Oct 13, 2026• Announced Oct 15, 2026• 0 citations
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Abstract

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 $\ln Δ^{(k)}\sim -σ^2$ at weak disorder and $\ln Δ^{(k)}\sim -1/σ$ at strong disorder, whose crossover sets an optimal disorder strength $σ_c\sim J_{xy}$, where $J_{xy}$ 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.

Comments
main text: 5 pages, 3 figures. Supplementary material 9 pages, 3 figures

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Optimal Disorder Strength for Approximate Unitary kkk-Designs in Spin Chains
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 kkk-design, which has so far been identified mainly empirically. Here, we develop a perturbative theory for the moment gap Δ(k)Δ^{(k)}Δ(k), the deviation of the kkk-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 ln⁡Δ(k)∼−σ2\ln Δ^{(k)}\sim -σ^2lnΔ(k)∼−σ2 at weak disorder and ln⁡Δ(k)∼−1/σ\ln Δ^{(k)}\sim -1/σlnΔ(k)∼−1/σ at strong disorder, whose crossover sets an optimal disorder strength σc∼Jxyσ_c\sim J_{xy}σc​∼Jxy​, where JxyJ_{xy}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 kkk-designs provide a resource-efficient framework for emulating Haar randomness up to the kkk-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 kkk-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.
The question of how well a finite ensemble of quantum states can approximate Haar randomness is relevant in many areas of quantum physics. Here we consider this problem from a geometrical perspective, via the notion of Welch bounds. The latter are a set of inequalities, originally derived in the context of signal processing, which precisely characterize exact k-designs, i.e. a finite ensemble of states that exactly reproduces the k-th moment of the Haar measure. We derive stronger Welch bounds which enable a quantitative and operationally meaningful characterisation of approximate k-designs. Our new bounds are often tight, allowing us to characterize optimal approximate designs in regimes where the cardinality of the ensemble is small. A notable example is complete sets of mutually unbiased bases (MUBs), which form an optimal approximate 3-design among 1-designs of their cardinality. This leads to a new variational approach to determine the existence of complete sets of MUBs, which we explore numerically in dimension six. We highlight a connection to shadow tomography, relating our bounds to the shadow norm. We comment on how the method extends to other types of designs, including toric and unitary designs. A key technical ingredient, is the computation of the complete spectrum of the partially transposed symmetric-subspace projector, including multiplicities and eigenvectors, which may find further applications.
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.
We present a single-quench protocol that generates unitary kkk-designs with minimal control. A system first evolves under a random Hamiltonian H1H_1H1​; at a switch time ts≥tTht_s \geq t_{\mathrm{Th}}ts​≥tTh​ (the Thouless time), it is quenched to an independently drawn H2H_2H2​ from the same ensemble and then evolves under H2H_2H2​. 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 kkk-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 tTht_{\mathrm{Th}}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.
Disordered one-dimensional interacting systems have long been characterized via conventional correlation functions. A complementary quantum-information perspective quantifies the randomness of the unitary ensemble dynamics generated by a quantum system through the frame potential, which serves as a practical diagnostic for quantum algorithmic performance. However, no analytical treatment has yet been achieved for experimentally accessible interacting one-dimensional systems. In this Letter, we derive a closed-form expression for the frame potential of a Tomonaga-Luttinger liquid with quenched Gaussian forward-scattering disorder. Exploiting the exactly quadratic structure of the disorder-averaged Keldysh action, we show that the frame potential decays as a power law at early times and saturates to a late-time plateau controlled by a single coupling parameter. Taking the random field XXZ spin chain as a specific microscopic realization, we show that the strongest randomness is achieved near the Heisenberg ferromagnetic point and can be exponentially enhanced through a multiple-quench protocol. We validate our results across the entire gapless phase, with direct implications for algorithm design in analog quantum simulation platforms.
Unitary designs provide finite-moment approximations to Haar-random unitaries, with wide-ranging applications across physics and quantum information, from scrambling and black-hole dynamics to foundational primitives in quantum algorithms. Strong unitary designs capture a more demanding operational notion of approximation, requiring indistinguishability from Haar randomness even for quantum algorithms that may access a unitary not only in the forward direction, but also through its inverse, transpose, and complex conjugate. Motivated by the physical requirement that scrambling arise within the system itself, Schuster, Ma, Lombardi, Brandão, and Huang (arXiv:2509.26310) left open whether strong unitary designs can be generated in logarithmic depth using only the system qubits. For every fixed design order kkk and measurable-error tolerance, we construct strong approximate unitary kkk-designs in optimal Θ(log⁡n)Θ(\log n)Θ(logn) all-to-all circuit depth using only the nnn original system qubits. Our new ingredient is a logarithmic-depth Pauli-mixing bound for the perfect-matching ensemble, whose layers pair the qubits uniformly at random and apply independent random two-qubit gates. This bound controls the mixed forward-reverse two-query case, which we combine with existing design and gluing results to obtain strong unitary designs of arbitrary fixed order.