Error-resilient Reversal of Quantum Chaotic Dynamics Enabled by Scramblons

TL;DR

Overall, the paper addresses reversing chaotic quantum dynamics in chaotic many-body systems with imperfect backward evolution and develops a scramblon-based framework to quantify and mitigate errors. They perform a macroscopic NMR experiment on adamantane, measure the OTOC via MQC, and fit the data with the scramblon ansatz $F_{\alpha\gamma}(\phi,t) = (1 + a e^{\varkappa t} + b \phi^2 e^{\varkappa t})^{-2\Delta}$ to extract a universal growth rate $\varkappa$. Mitigating backward errors by setting $a=0$ recovers the exponential growth regime of the OTOC, revealing the quantum Lyapunov exponent $\varkappa$ in a macroscopic system. These results validate scramblon theory in a realistic long-range interacting platform and suggest pathways for error-resilient quantum simulation and metrology.

Abstract

The emergence of the arrow of time in quantum many-body systems stems from the inherent tendency of Hamiltonian evolution to scramble quantum information and increase entanglement. While, in principle, one might counteract this temporal directionality by engineering a perfectly inverted Hamiltonian to reverse entanglement growth, such a scenario is fundamentally unstable because even minor imperfections in the backward evolution can be exponentially amplified, a hallmark of quantum many-body chaos. Therefore, successfully reversing quantum many-body dynamics demands a deep understanding of the underlying structure of quantum information scrambling and chaotic dynamics. In this letter, by using solid-state nuclear magnetic resonance on a macroscopic ensemble of randomly interacting spins, we measure the out-of-time-ordered correlator (OTOC) and validate key predictions of scramblon theory, a universal theoretical framework for information scrambling. Crucially, this theory enables us to isolate and mitigate errors in the OTOC caused by imperfections in the backward evolution. As a result, this protocol uncovers the anticipated exponential behavior of quantum many-body chaos and extracts the quantum Lyapunov exponent in a many-body experimental system for the first time. Our results push the fundamental limits of dynamical reversibility of complex quantum systems, with implications for quantum simulation and metrology.

Figures (4)

• Figure 1: Schematic of Experimental Protocol. (a) Chaotic quantum many-body dynamics: Unitary evolution under $\hat{H}$ scrambles quantum information, transforming a low-entanglement initial state into a highly entangled state. Under time-reversed dynamics with perturbed Hamiltonian $-\hat{H}+\delta\hat{H}$, the system exhibits exponential deviation from perfect state recovery. (b) Microscopic structure of adamantane (C$_{10}$H$_{16}$) powder: Each granule contains adamantane molecules arranged in a face-centered cubic lattice. Individual molecules consist of spin-$1/2$$^1$H nuclei and spinless $^{12}$C atoms. (c) Top: Floquet pulse sequence converting the dipolar Hamiltonian Eq. \ref{['dipolar']} into the engineered form Eq. \ref{['Heff']}suter1987multipleCappellaroExploringLocalizationNuclear2018a. Bottom: Experimental sequence for MQC and OTOC measurements.
• Figure 2: Fitting Experimental Data with the Scramblon Ansatz. Three typical experimental data $F^{(\lambda)}_{\alpha\gamma}(\phi,t)$ for three different $\phi=0$, $\pi/64$ and $\pi/32$. The superscript $\lambda$ labels three different Hamiltonians, as described in the main text. The solid curves are fitting results obtained using the scramblon ansatz. Experimentally, the $95\%$ confidence intervals determined from read-out noise are approximately $10^{-4}$, and therefore the error bars are contained within the data markers.
• Figure 3: Verifying the Predictions of Scramblon Theory. The three columns correspond to the fitting parameters extracted from three different Hamiltonians: $\hat{H}^{(1)}$, $\hat{H}^{(2)}$, and $\hat{H}^{(3)}$. Subfigures (a)-(c) in the upper row display the fitting parameters for different values of $\alpha$ and $\gamma$, with green diamonds representing $a\times50$, yellow squares representing $\Delta$, and blue triangles representing $\varkappa$. The dashed lines represent $\varkappa$ averaged over $\alpha$ and $\gamma$, and $a$ and $\Delta$ averaged over $\alpha$. Subfigures (d)-(f) in the lower row show the eigenvalues $\sigma_1,\sigma_2,\sigma_3$ for the symmetric matrix $M^\text{s}$ and the absolute value of eigenvalues $\sigma_4,\sigma_5,\sigma_6$ for the antisymmetric matrix $M^\text{a}$. The error bars on the fitting parameters represent the $95\%$ confidence interval, including the standard deviation from each fitting residual $\sigma_{\text{res}}$ and the uncertainty arising from the choice of fitting range $\sigma_{\text{range}}$SM.
• Figure 4: Error Mitigation of the Reversed Quantum Many-Body Dynamics. (a) $F_{\alpha\gamma}(\phi, t)$ and (b) $I_{\alpha\gamma}(t)$ defined in Eq. \ref{['eq:OTOC_commutator']}. As an example, we show $\alpha,\gamma=z,y$, $\phi=\pi/64$ and with Hamiltonian $\hat{H}^{(1)}_0$. The blue solid line results from the scramblon ansatz by setting $a=0$. The red triangles are error mitigated by $\tilde{F}_{\alpha\gamma}(\phi,t)$, and green circles are raw data without error mitigation. The red dash-dotted line and the green dashed line are obtained from the scramblon ansatze of $\tilde{F}_{\alpha\gamma}(\phi,t)$ and $F_{\alpha\gamma}(\phi,t)$ (without setting $a=0$). The experimental data points in (b) were extracted by fitting the corresponding data for $F_{\alpha\gamma}(\phi,t)$ with a sixth-order polynomial. The errorbars represent the standard deviations.