Edge of Many-Body Quantum Chaos in Quantum Reservoir Computing

TL;DR

This work extends reservoir computing into quantum many-body systems using the SYK model to identify two edges of quantum chaos that optimize QRC performance: a temporal edge set by the Thouless time $t_{\mathrm{Th}}$ and a parametric edge at the chaotic–integrable transition. By analyzing level statistics and the spectral form factor $K(t)$, the authors connect chaos diagnostics to practical learning performance on STM and NARMA tasks, showing maximal expressivity near these edges and a Haar baseline beyond $t_{\mathrm{Th}}$. The temporal edge yields peak enhancements for chaotic SYK$_4$, while the parametric edge appears in the interpolated model under long input intervals, illustrating how memory and nonlinearity requirements determine the optimal operating regime. These results provide a design guideline for quantum reservoirs, suggesting that tuning to the edges of many-body quantum chaos can enhance information processing in quantum devices with complex dynamics.

Abstract

Reservoir computing (RC) is a machine learning paradigm that harnesses dynamical systems as computational resources. In its quantum extension -- quantum reservoir computing (QRC) -- these principles are applied to quantum systems, whose rich dynamics broadens the landscape of information processing. In classical RC, optimal performance is typically achieved at the ``edge of chaos," the boundary between order and chaos. Here, we identify its quantum many-body counterpart using the QRC implemented on the celebrated Sachdev-Ye-Kitaev model. Our analysis reveals substantial performance enhancements near two distinct characteristic ``edges": a temporal boundary defined by the Thouless time, beyond which system dynamics is described by random matrix theory, and a parametric boundary governing the transition from integrable to chaotic regimes. These findings establish the ``edge of many-body quantum chaos" as a design guideline for QRC.

Figures (16)

• Figure 1: Schematic illustration of the SYK model and its implementation in QRC architecture. The SYK system described in Eq. (\ref{['eq1']}), which serves as the quantum reservoir, comprises the SYK$_4$ term ($J_{ijkl}$) and the SYK$_2$ term ($\kappa_{ij}$). At each time step $k$, the process begins by encoding an input $u^{(k)}$ into the quantum reservoir via the input state $\rho_{\mathrm{in}}^{(k)}$. The reservoir state $\rho^{(k)}$ then evolves under the Hamiltonian according to the update rule in Eq. (\ref{['eq2']}). Information is subsequently read out by constructing a state vector $\bm{\mathrm{x}}^{(k)}$ from the expectation values $\langle c_i^\dagger c_i\rangle$ and a constant bias term. Finally, this vector is linearly transformed by the trainable weight matrix $W_{\mathrm{out}}$ to produce the output $y^{(k)}$, which is optimized to approximate the target $\bar{y}^{(k)}$.
• Figure 2: (a) Distribution of level spacing ratio $P(r)$ for the SYK$_4$ model (blue) and the SYK$_2$ model (red). Corresponding WD and Poisson predictions are shown in blue and red curves, respectively. (b) Level spacing ratio $\langle r \rangle$ as a function of the coupling strength $\kappa_2/J_4$, computed over the central $50\%$ of the spectrum. The horizontal dashed lines represent the reference values for Poisson and WD statistics. In both panels, data are averaged over $2000$ realizations for the system size $N=8$.
• Figure 3: (a), (b) The SFF $K(t)$ for the SYK$_4$ and SYK$_2$ models, with each curve averaged over $20000$ realizations. (c), (d) The QRC performances as functions of $\Delta t_{\mathrm{in}}$ for the SYK$_4$ and SYK$_2$ models. The upper panels show the memory performance $R^2_{d=n-1}$ (higher values indicate better retention), while the lower panels display the NMSE on the NARMA tasks (lower values indicate better accuracy). Markers represent the orders $n=2,3,5$, and $7$ (plotted sparsely for brevity). The horizontal dotted lines indicate the reference performance of the Haar QRC model, with poorer performance corresponding to higher $n$. Performance results are averaged over $500$ realizations, with shaded regions denoting the standard deviation. Vertical dashed lines in (a) and (c) represent the Thouless time $t_{\mathrm{Th}}$.
• Figure 4: (a), (b) The QRC performance as a function of $\kappa_2 / J_4$ for the input interval $\Delta t_{\mathrm{in}}=50$ and $\Delta t_{\mathrm{in}}=1$, respectively, analogous to Figs. \ref{['fig3']}(c) and \ref{['fig3']}(d). Sparsely placed markers denote the order $n$, with colors indicating $\langle r \rangle$ according to the color scale in Fig. \ref{['fig2']}(b). The vertical dashed lines indicate the approximate boundaries where the absolute difference between the measured $\langle r\rangle$ and its theoretical value for WD (left) and Poisson (right) statistics first falls below $10^{-2}$.
• Figure S1: (a), (b) Distribution of level spacing ratio $P(r)$ for the Hamiltonian including the PHS preserving term: $\mathcal{H} + \mathcal{H}^{\mathrm{PHS}}$. Panel (a) corresponds to a system size of $N=8$, while panel (b) uses $N=7$. In both cases, the particle number sector is fixed at $N_p = \lfloor N/2\rfloor$. Theoretical predictions for Poisson (red), GOE (green), and GUE (blue) statistics are shown as curves. The histograms present $P(r)$ for the SYK$_2$ model (red) and the SYK$_4$ model [green for (a) and blue for (b)]. Data are averaged over $2000$ independent realizations in both panels.