Practical Few-Atom Quantum Reservoir Computing

TL;DR

This work addresses building a practical quantum reservoir computer using a few two-level atoms in an optical cavity to process temporal data with memory and nonlinearity. It couples atoms through a common cavity mode and extracts features via continuous non-destructive measurements, incorporating polynomial post-processing to boost expressivity. The Mackey-Glass forecasting task and sine-square waveform classification demonstrate that increasing the atom count dramatically expands the reservoir's Hilbert space (scaling as $2^{N_{atom}}$) and improves performance beyond classical echo-state networks, even under realistic measurement conditions. The approach offers a scalable, energy-efficient hardware path for quantum-inspired temporal computation and highlights the role of hardware heterogeneity in preserving performance.

Abstract

Quantum Reservoir Computing (QRC) harnesses quantum systems to tackle intricate computational problems with exceptional efficiency and minimized energy usage. This paper presents a QRC framework that utilizes a minimalistic quantum reservoir, consisting of only a few two-level atoms within an optical cavity. The system is inherently scalable, as newly added atoms automatically couple with the existing ones through the shared cavity field. We demonstrate that the quantum reservoir outperforms traditional classical reservoir computing in both memory retention and nonlinear data processing through two tasks, namely the prediction of time-series data using the Mackey-Glass task and the classification of sine-square waveforms. Our results show significant performance improvements with an increasing number of atoms, facilitated by non-destructive, continuous quantum measurements and polynomial regression techniques. These findings confirm the potential of QRC as a practical and efficient solution to addressing complex computational challenges in quantum machine learning.

Figures (11)

• Figure 1: Setup of quantum optical reservoir computing. The reservoir is composed of atoms inside an optical cavity, exhibiting diverse detunings and couplings across various spatial positions. The input function is integrated into the coherent driving of the cavity. Feature vector is obtained via continuous measurements of photonic quadratures and atomic spin channels. A machine learning process is employed to train the mapping from the feature vector to the output.
• Figure 2: Input and features for the Mackey-Glass task. (a) The input function, $f_{k}$, divided into memory fading, training, and testing zones. (b)-(e) The corresponding feature from a single-atom reservoir with $\omega_{1}=20$ and $g_{1}=30$. Parameters: $\tau=20$, $\kappa=10$, $\omega_{c}=40$, and $\epsilon=20$.
• Figure 3: Testing result for the Mackey-Glass task with various reservoir scales. (a) NRMSE plotted against the number of features, where features are obtained from all atoms and the photon mode. Blue lines: fixed $g_{i}=30$ for all atoms and $\omega_{i}=20$ for one atom, $\omega_{i}=[0,40]$ for two atoms, $\omega_{i}=[0,20,40]$ for three atoms, $\omega_{i}=[0,10,30,40]$ for four atoms, and $\omega_{i}=[0,10,20,30,40]$ for five atoms. Green lines: fixed $\omega_{i}=20$ for all atoms and $g_{i}=30$ for one atom, $g_{i}=[10,50]$ for two atoms, $g_{i}=[10,30,50]$ for three atoms, $g_{i}=[10,20,40,50]$ for four atoms, and $g_{i}=[10,20,30,40,50]$ for five atoms. Solid lines: regular linear regression. Dashed lines: polynomial regression incorporating both linear and quadratic terms of observables. Red line: CRC averaging $1000$ random trajectories on the echo state network. (b) NRMSE as a function of the number of hidden (unmeasured) atoms in QRC, while maintaining measurement of $4$ specified observables from the cavity field and the particular atom with $g_{1}=30$ and $\omega_{1}=20$. (c)-(d) Actual (red) and target (blue) outputs from QRC with one atom (linear regression) and five atoms (polynomial regression), corresponding to the points marked by letters "c" and "d" in panel (a), respectively. Parameters: $\tau=20$, $\kappa=10$, $\omega_{c}=40$, $\epsilon=20$.
• Figure 4: Testing results for the Mackey-Glass task within the framework of continuous quantum measurement, where each observable feature is obtained by averaging measurement records over multiple measurement trajectories. (a) NRMSE as a function of the number of measurement trajectories for a three-atom QRC with $g=30$ and $\omega_{i}=[0,20,40]$. Blue lines: averaged results from a finite number of trajectories simulated with stochastic master equation in Eq. (\ref{['eq:sme']}). Orange lines: asymptotic results from infinite number of trajectories simulated with deterministic master equation in Eq. (\ref{['eq:me']}). Solid lines: regular regression. Dashed lines: polynomial regression. (b)(c) The actual (red) and target (blue) outputs for $10$ and $10000$ trajectories, corresponding to the points labeled "b" and "c" in panel (a), respectively. Parameters: $\tau=20$, $\kappa=10$, $\omega_{c}=40$, $\epsilon=20$, $dt=1$.
• Figure 5: Testing result for the Mackey-Glass task with various delay $\tau$. (a) NRMSE as a function of delay $\tau$ for a three-atom QRC with $g=30$ and $\omega_{i}=[0,20,40]$. (b)(c) The actual (red) and target (blue) outputs with $\tau=2$ and $\tau=200$, corresponding to the points marked by letters "b" and "c" in panel (a), respectively. Parameters: $\kappa=10$, $\omega_{c}=40$, $\epsilon=20$.