High-Accuracy Temporal Prediction via Experimental Quantum Reservoir Computing in Correlated Spins

Abstract

Physical reservoir computing provides a powerful machine learning paradigm that exploits nonlinear physical dynamics for efficient information processing. By incorporating quantum effects, quantum reservoir computing offers superior potential for machine learning applications, as quantum dynamics are exponentially costly to simulate classically. Here, we present a novel quantum reservoir computing approach based on correlated quantum spin systems, exploiting natural quantum many-body interactions to generate reservoir dynamics, thereby circumventing the practical challenges of deep quantum circuits. Our experimental implementation supports nontrivial quantum entanglement and exhibits sufficient dynamical complexity for high-performance machine learning. We achieve state-of-the-art performance in experiments on standard time-series benchmarks, reducing prediction error by 1 to 2 orders of magnitude compared to previous quantum reservoir experiments. In long-term weather forecasting, our 9-spin quantum reservoir delivers greater prediction accuracy than classical reservoirs with thousands of nodes. This represents the first experimental demonstration of quantum machine learning outperforming large-scale classical models on real-world tasks.

Figures (12)

• Figure 1: Schematic diagram of quantum reservoir computing. The input series $\{s_k\}$ is mapped to the control parameters $\{\theta_k\}$ of a sequence of quantum operations, which are sequentially applied to a quantum reservoir comprising a spin network. The reservoir state is probed through measurements of partial observables, producing a readout vector with components $x^{i}_k = \langle O^i \rangle_k = {\rm Tr}(\rho_k O^i)$. The final output is computed as a linear combination, $\hat{y}_k={\bf x}_{k}^{\top} {\bf w}$.
• Figure 2: Schematic of experimental setup and readout schemes. (a) Circuit diagram of the QRC experimental implementation. The reservoir, composed of carbon and proton spins, undergoes repeated cycles of input-dependent rotations $\{\phi_k,\theta_k\}$ followed by intrinsic evolution under $\exp(\tau\mathcal{L}_H)$. The experiment adopts a rewinding protocol, where each experiment at time step $k$ is restarted from cycle $k-n_{\rm wo}$ to wash out initial state dependence, and readout is performed on the proton spins at the end of the circuit. Two readout schemes are considered. (b) Single-time readout: analogous to capturing multiple images of a static system by artificially changing the viewpoint. A fixed set of observables $\{O^i\}$ (e.g. Pauli operators) is measured on the reservoir state $\rho_k$, yielding the readout vector ${\bf x}_k = [\langle O^1\rangle_k, \langle O^2\rangle_k, \dots]^\top$ where $\langle O\rangle_k={\rm Tr}(\rho_k O)$. (c) Time-multiplexed readout: analogous to filming a moving system from a fixed viewpoint, where other aspects naturally rotate into view. A single observable $O$ is sampled at multiple time points during the reservoir's evolution, equivalently measuring the Heisenberg‑picture operators $\{O(t_i)\}$, yielding the readout vector ${\bf x}_k = [\langle O(t_1)\rangle_k, \langle O(t_2)\rangle_k, \dots]^\top$.
• Figure 3: High accuracy temporal prediction with NARMA benchmark.(a) Prediction performance (evaluated by $R^2$) on NARMA tasks with $n$ varying from 2 to 20, using quantum reservoirs with different configurations. Red and blue lines represent evolution durations of $\tau = 0.3s$ and $0.01s$, respectively, while solid and dashed lines indicate results obtained with time-multiplexed (TM) and single-time readout schemes. (b) Prediction results for NARMA tasks with $n=10$ and $20$, utilizing time-multiplexed readout with $\tau=0.01s$.
• Figure 4: Weather forecasting via quantum reservoir computing. (a) Experimental results for single-step-ahead forecasting of temperature and humidity using quantum reservoir computing. The black and red lines denote the target and predicted trajectories, respectively. (b) Multi-step-ahead forecasting performance comparison between echo state networks (ESNs) with varying node counts and QRC methods. $R^2$ is averaged over 100 random realizations for each ESN($m$), with shaded regions representing the standard deviation as error bars.
• Figure 5: Feature extraction from the NMR free-induction-decay (FID) signal. (a) The real part of the FID signal obtained from the proton spins of crotonic acid, showing the first 2048 points (one quarter of the total acquisition). The signal amplitude is enhanced by the spectrometer's receiver gain. (b) The real part of the corresponding NMR frequency spectrum, produced by Fourier transforming the FID signal. The intense peak near 1880 Hz originates from $\rm H_2O$ in the solvent. The red, green, and orange boxes highlight the spectral regions of crotonic acid's $\rm H_1$, $\rm H_2$, and methyl protons $\rm H_{Me}$ (i.e., the equivalent $\rm H_3, H_4, H_5$), with insets showing magnified views. Although these signal intensities are two to three orders of magnitude lower than that of the $\rm H_2O$ peak, they still exhibit sufficiently high signal-to-noise ratios. (c) Normalized readout vector constructed by concatenating the spectral amplitudes from the highlighted regions, serving as time-multiplexed readout features for QRC.