The Role of Entanglement in Quantum Reservoir Computing with Coupled Kerr Nonlinear Oscillators

TL;DR

The paper investigates whether entanglement in a quantum reservoir computing system built from two coupled Kerr nonlinear oscillators enhances time-series forecasting. Using Hamiltonian encoding with $H(t)=H_{nl}+H_{int}+H_{drive}$ and open-system Lindblad dynamics, they evaluate linear and nonlinear tasks with time-multiplexed Fock-basis readouts and ridge regression, quantifying entanglement via $E_N$ and performance via NRMSE. They uncover a frequency-threshold regime in which moderate entanglement correlates with lower average and worst-case errors, a relation strengthened by sufficient dissipation but diminished by dephasing; this trend persists in nonlinear (NARMA-20) tasks. These findings suggest entanglement is a relevant quantum resource for reservoir-based learning under realistic noise and dimensional constraints, motivating future work on larger oscillator networks and alternative entanglement measures to generalize the observed advantage.

Abstract

Quantum Reservoir Computing (QRC) uses quantum dynamics to efficiently process temporal data. In this work, we investigate a QRC framework based on two coupled Kerr nonlinear oscillators, a system well-suited for time-series prediction tasks due to its complex nonlinear interactions and potentially high-dimensional state space. We explore how its performance in forecasting both linear and nonlinear time-series depends on key physical parameters: input drive strength, Kerr nonlinearity, and oscillator coupling, and analyze the role of entanglement in improving the reservoir's computational performance, focusing on its effect on predicting non-trivial time series. Using logarithmic negativity to quantify entanglement and normalized root mean square error (NRMSE) to evaluate predictive accuracy, our results suggest that entanglement provides a computational advantage on average -- up to a threshold in the input frequency -- that persists under some levels of dissipation and dephasing. In particular, we find that higher dissipation rates can enhance performance. While the entanglement advantage manifests as improvements in both average and worst-case performance, it does not lead to improvements in the best-case error. These findings contribute to the broader understanding of quantum reservoirs for high performance, efficient quantum machine learning and time-series forecasting.

Figures (17)

• Figure 1: Overview of the coupled Kerr Oscillator QRC. The oscillators are driven by the input signal with amplitudes $\epsilon_a$ and $\epsilon_b$ at a coupling strength of $g$. The reservoir state is measured in the Fock basis (read-out) at moments of $t=\delta t/m$ at a constant rate (sampling via multiplexing). In response, the quantum state of the system evolves, and the computational nodes evolve due to the natural dynamics of the system and input signal. The output of the reservoir is fed to the ridge regression to generate weights. The weights are then used to generate the output time series, which is a prediction of the target.
• Figure 2: Time evolution of entanglement (top row) and corresponding time-series prediction performance (bottom row) at different input frequencies $f$. Top: Behavior of logarithmic negativity in time during testing and training. To ensure the reservoir reflects stable dynamics rather than initial transients, we use the portion of the evolution after the vertical dashed line for training and testing. Bottom: Examples of input sequences at four different frequency scales and The corresponding predictions of the reservoir at coupling rate of $g=0.9$ , input strengths of $\epsilon_a=\epsilon_b=3$, nonlinearities of $K_a=K_b=1$ and dissipation rates of $\kappa_a=\kappa_b=0.1$ in a rotating frame. The performance gets worse as the input frequency increases.
• Figure 3: NRMSE (left) and entanglement (right) versus input strength $\epsilon$ for different input frequencies, with fixed parameters $K = 1$, $\kappa_a = \kappa_b = 0.1$, $g=0.9$, and $\kappa_\phi = 0$. Left: While the first dips are observed at $\epsilon \simeq 2$, NRMSE decreases with increasing input strength up to $\epsilon \simeq 5$, after which it saturates or worsens depending on the input frequency. Right: Logarithmic negativity increases with $\epsilon$ until a peak near $\epsilon \simeq 2$, beyond which it declines with a lower slope. Best-case errors correspond to moderate entanglement for frequencies below the threshold (e.g., $f=2.5$).
• Figure 4: NRMSE (left) and Entanglement (right) versus Kerr nonlinearity $K$ for different input frequencies, with fixed parameters $\epsilon = 3$, $\kappa_a = \kappa_b = 0.1$, $g=0.9$, and $\kappa_\phi = 0$. Left: NRMSE fluctuates up to $K \simeq 2$ for $f = 0.8$ and $f = 1.5$, and increases thereafter. For $f=0.05$, it increases monotonically, while no clear trend is observed for $f = 2.5$. Right: Entanglement fluctuates up to $K \simeq 1$, beyond which it decreases. Best-case errors correspond to moderate entanglement for frequencies below the threshold (e.g., $f=2.5$).
• Figure 5: NRMSE (left) and Entanglement (right) versus coupling constant $g$ for different input frequencies, with fixed parameters $\epsilon = 3$, $\kappa_a = \kappa_b = 0.1$, $K=1$, and $\kappa_\phi = 0$. Left: NRMSE for $f = 0.8$ and $f = 1.5$ fluctuates up to $g \simeq 1.4$, beyond which it increases. For $f=0.05$, NRMSE increases monotonically, and no clear trend is observed for $f = 2.5$. Right: Entanglement peaks near $g \simeq 0.9$. Beyond this point, entanglement decreases. Best-case errors (third points from the left on the left panel) correspond to moderate entanglement (the same points for the right panel) for frequencies below the threshold (e.g., $f=2.5$).