Modular quantum extreme reservoir computing

TL;DR

This work addresses how to design modular quantum reservoirs that match a single large reservoir with minimal inter-module wiring. By explicitly separating intra-module dynamics from inter-module couplings, it develops the MQERC framework and analyzes three inter-module connectivity schemes across two- and three-module architectures, using ZZ-type interactions and distance-dependent couplings, with entanglement entropy $\overline{S}$ as a key diagnostic. The results show that a handful of well-placed inter-module connections, particularly one-to-one parallel links, can reproduce or closely approach the performance of a full reservoir on MNIST, Fashion-MNIST, and CIFAR-10, with a positive correlation between inter-module entanglement and accuracy. The findings generalize to random modular reservoirs and scale to three modules, underscoring the practical potential for hardware-friendly modular quantum reservoir designs on two-dimensional chips or distributed small quantum systems, and suggesting applicability to time-series tasks as well.

Abstract

Quantum reservoir computing employs fixed quantum dynamics as a feature map for machine learning. Integrating multiple quantum reservoirs, however, raises a key question: how few inter-module connections are sufficient to match the performance of a single reservoir? To address this, we explicitly separate intra-module dynamics from inter-module couplings and systematically examine different connectivity schemes. We find that even a small number of well-placed connections between two modules can match single-reservoir accuracy, with simple one-to-one connections proving highly effective. Performance generally improves with increasing inter-module entanglement, and these correlations persist for both $ZZ$-coupled and random modular reservoirs. Extensions to three modules and evaluations across multiple datasets (MNIST, Fashion-MNIST, CIFAR-10) suggest that the modular architecture can be applied to diverse reservoir types and image-classification datasets. These results motivate modular quantum reservoir designs that align naturally with realistic hardware, such as two-dimensional quantum-chip layouts or networks of small integrated quantum systems.

Figures (7)

• Figure 1: Schematic illustration of MQERC and its architecture. (a) MQERC data flow involving classical pre-processing, a quantum reservoir, and a trainable classical linear classifier. (b) Quantum reservoir structure consisting of encoder $\mathcal{U}_{\mathrm{enc}}$ encoding processed classical data $\mathbf{I}^{(s)}$ into quantum states, module-level reservoirs $\mathcal{U}_{\mathrm{M}}$, inter-module coupling $\mathcal{U}_c$, single-qubit rotations $\mathcal{R}_x$, and a measurement channel $\mathcal{M}$ resulting in the output $\mathbf{p}^{(s)}$. (c) Different inter-modular connectivity structures. $R_\times$ is the boundary range, $n_\times$ counts boundary connections, $n_a$ counts additional arbitrary inter-module connections outside the $R_\times$ set, $n_\ell$ counts the parallel connections. See the main text for details.
• Figure 2: Test accuracy $\eta$ for the image classification tasks using the (a) MNIST, (b) Fashion-MNIST, (c) CIFAR-10 datasets on a single 10-qubit chain vs interaction exponent $\alpha$ for varying interaction range $R$ with $\theta_J=2\pi$ and $\theta_g=\pi/8$. Further we show for the (d) MNIST, (e) Fashion-MNIST, (f) CIFAR-10 datasets, $\eta$ vs $\theta_J$ at $\alpha=1.5$ for varying $R$.
• Figure 3: Test accuracy $\eta$ vs inter-modular interaction strength $\theta_c$. Each curve represents a cross range $R_{\times}=1,2,3,4$ (corresponding to the number of cross connections $n_\times=1,3,6,10$). The horizontal dashed lines are 10-qubit single-chains with interaction range $R$ for comparison. Two modules with a [5,5]-module structure, with $\theta_J=2\pi$, $\theta_g=\pi/8$, and $\alpha=1.5$.
• Figure 4: Highest test accuracy $\eta^*$ over all possible configurations vs number of one-to-one parallel inter-modular connections $n_\ell$ for different datasets. (a-c) [5,5]-module with 32 possible parallel-connection configurations. Small pink circles: $\eta$ with $\theta_c=\pi/4$ for each configuration. Pink triangles: highest $\eta^*$ over configurations at fixed $n_\ell$ and $\theta_c=\pi/4$. Red circles: highest $\eta^*$ over configurations and $\theta_c$ at fixed $n_\ell$. Black circles: For arbitrary connections, positioned at $n_\ell = n_a$ on the x-axis, highest $\eta^*$ over configurations at fixed $n_a$ and $\theta_c=\pi/4$. (d-f) [5,5,5]-module with 1024 possible configurations. Connected: Both neighboring module pairs have at least one connection. Disconnected: Otherwise. Small circles and triangles: Same meaning as in (a-c). Dashed gray lines: Baselines of single chains with the same total qubit count. Insets: Possible inter-module connections are illustrated by dotted lines. $\theta_J=2\pi$, $\theta_g=\pi/8$, and $\alpha=1.5$. See the main text for details.
• Figure 5: Inter-module entanglement entropy $\overline{S}$ vs $\theta_c$ for MNIST. $\overline{S}$ is the entanglement entropy between the two modules in the final states, averaged over the test set, with a shaded area indicating one standard deviation. (a) Boundary-crossing connections $R_{\times}$ corresponding to Fig. \ref{['fig:2module_thetac']}. (b) One-to-one parallel connections $n_\ell$ corresponding to Fig. \ref{['fig:mqerc_parallel_acc']}a, with one specific configuration plotted for each $n_\ell$. A "1" in a configuration indicates the presence of a one-to-one parallel connection at the corresponding location for the [5,5]-module.