Unified Probe of Quantum Chaos and Ergodicity from Hamiltonian Learning

Abstract

Developing measures of quantum ergodicity and chaos stands as a foundational task in the study of quantum many-body systems. In this work, we propose metrics for these effects based on Hamiltonian learning that unify multiple advantages of existing metrics. In particular, we show how ergodicity and chaos improve the robustness of Hamiltonian learning to small errors and furthermore demonstrate that this robustness can be used as a metric for such phenomena. We analytically and numerically show that our metrics not only distinguish between integrable and ergodic regimes in various spin chains but also quantify chaos and ergodicity, allowing us to locate regions of parameter space displaying maximal ergodicity and maximal sensitivity to local perturbations. Our approach not only provides conceptual ways to study quantum chaos and ergodicity but also presents viable experimental methods for quantum simulators.

Figures (8)

• Figure 1: Ergodicity and Chaos from Hamiltonian Learning. (a) We outline the general procedure for extracting our metrics for ergodicity and chaos. After preparing an (approximate) infinite-temperature eigenstate $\ket{v}$ of a Hamiltonian $H$, local basis measurements are used via e.g. shadow tomography Huang_2020 to construct the covariance matrix $M_{\alpha \beta}$ [Eq. \ref{['eq:cov matrix']}]. The eigenvalues $\sigma_a^2$ of $M$ constitute the variance spectrum of $\ket{v}$. $H$ is learned from the zero-eigenvalue subspace of $M$, while the rest of the variance spectrum acts as an indicator for ergodicity and chaos. (b) In integrable systems, the variance spectrum shows a wide spread and a small gap. (c) In contrast, ergodic systems are characterized by a narrow variance spectrum with a large gap. (d) We use these metrics (the spread $D_E$ and gap $\Delta_M$) to quantify ergodicity, while the maximum eigenvalue $\sigma^2_{\mathrm{max}}$ probes chaos by quantifying the maximum sensitivity of the eigenstate to a set of local perturbations.
• Figure 2: Variance Spectra in Integrable and Ergodic Hamiltonians. We calculate the variance spectra for all eigenstates of the transverse field Ising model in (a) [Eq. \ref{['eq:Ising Hamiltonian']}, $g = 1$, $h = 0$], the Heisenberg model in (b) [Eq. \ref{['eq:XXZ Hamiltonian']}, $\Delta = 1$], and the mixed field Ising model in (c) [Eq. \ref{['eq:Ising Hamiltonian']}, $g = 1$, $h = 0.3$] at system size $N=12$. The Hamiltonian is always learned as a zero-variance operator (orange), whereas the rest of the spectra (green) show a clear qualitative difference between integrable and ergodic models. We calculate the inverse spread $1/D_E$ [Eq. \ref{['eq:D_E definition']}] in (d) and the gap $\Delta_M$ [Eq. \ref{['eq:Delta_M definition']}] in (e) of the spectra for each model averaged over a microcanonical distribution of eigenstates at energy $E = 0$ (the ribbons denote the standard deviation of each metric over the ensemble). The inset in (d) plots $1/D_E$ on a log scale to show exponential growth. The inverse spread grows parametrically faster with system size $N$ in the ergodic model than in the integrable ones, and the variance gap similarly increases more quickly. We derive this parametric difference between the ergodic and free fermion (TFIM) models in Appendices \ref{['app:rmt predictions']} and \ref{['app:ff predictions']}.
• Figure 3: Concentration and Anti-Concentration of Variance Spectra. (a) By modeling ergodic midspectrum eigenstates as Haar random, we derive an upper bound [Eq. \ref{['eq:levy bound for var spectrum']}] on the probability that any nonzero variance $\sigma_a^2 \neq 0$ is more than $\varepsilon$ away from 1. This probability decays like an exponential in the Hilbert space dimension $d$, demonstrating the robustness of Hamiltonian learning in the ergodic regime. (b) For translationally invariant free fermion systems, we derive an (approximate) lower bound [Eq. \ref{['eq:ff anti-concentration']}], consistent in the limit of large $N$, on the probability that any nonzero variance is smaller than $1-\varepsilon$ for an eigenstate selected uniformly at random. This probability decays like a power law in $d$, showing that Hamiltonian learning is parametrically less robust in this class of integrable systems than in ergodic systems. This bound is not constructed from conserved quantities; rather, we multiply the Hamiltonian by a sinusoidal spatial profile to construct an operator with low variance for an exponential number of eigenstates [see Appendix \ref{['app:ff predictions']}].
• Figure 4: Ergodicity Metrics in the Mixed Field Ising Model. Using the $N=14$ mixed field Ising model, we compare established ergodicity metrics to our proposed metrics for a microcanonical distribution of $600$ eigenstates centered at $E=0$. (a) The average level spacing $\langle r \rangle$ [Eq. \ref{['eq:level spacing ratio']}] is able to distinguish an ergodic region of parameter space (dark red) from the two integrable lines $g=0$ and $h=0$. (b) In contrast, the entanglement statistics measure $D_{\rm KL}$ [Eq. \ref{['eq:KL divergence definition']}] from Rodriguez-Nieva_2024 is not only able to distinguish integrable from ergodic but also reveals structure within the ergodic regime. As such, it is able to locate a pocket of maximal ergodicity near $g = 1$, $h = 0.3$, where agreement with the predictions of RMT is maximized. We then plot our proposed metrics for ergodicity: the variance gap $\Delta_M$ (c) and inverse spread $1/D_E$ (d). Both metrics are able to distinguish between integrable and ergodic regimes and identify the maximally ergodic region originally found with $D_{\rm KL}$. The blank regions in (a,b) near $g=0$ come from divergences and numerical instabilities in the metrics for classically integrable systems.
• Figure 5: Probing Chaos via Eigenstate Sensitivity. We compute the average maximal variance of the variance spectra from Fig. \ref{['fig:ergodicity maps']} as a probe of the sensitivity of eigenstates to local perturbations. (a) This metric identifies a window of parameter space near $g \to 0$ for $N = 14$ where eigenstates have high sensitivity, separating integrable and ergodic limits Pandey_2020LeBlond_2021Kim_2025 despite looking nearly integrable in Fig. \ref{['fig:ergodicity maps']}. (b) If we furthermore consider a slice of this parameter space ($h=0.3$), we see a transient increase of $\sigma_{\mathrm{max}}^2$ with system size in the sensitive region before it shrinks as the ergodic region expands. (c) In contrast, we observe no such scaling between the ergodic region and the transverse field limit ($g=1$). We attribute this lack of sensitivity to the strict locality of the operators $\mathcal{O}_a$. Ribbons denote the standard deviation of $\sigma^2_{\mathrm{max}}$ over the ensemble of states.