Quantum reservoir computing for predicting and characterizing chaotic maps

Abstract

Quantum reservoir computing has emerged as a promising paradigm for harnessing quantum systems to process temporal data efficiently by bypassing the costly training of gradient-based learning methods. Here, we demonstrate the capability of this approach to predict and characterize chaotic dynamics in discrete nonlinear maps, exemplified through the logistic and Hénon maps. While achieving excellent predictive accuracy, we also demonstrate the optimization of training hyperparameters of the quantum reservoir based on the properties of the underlying map, thus paving the way for efficient forecasting with other discrete and continuous time-series data. Using closed-loop prediction of distant future steps, our protocol discriminates between chaotic and nonchaotic phases without prior knowledge of the underlying map or the nature of the time series. Furthermore, the framework exhibits robustness against decoherence when trained in situ and shows insensitivity to reservoir Hamiltonian variations as well as robustness to finite-sampling error. These results highlight quantum reservoir computing as a scalable and noise-resilient tool for modeling complex dynamical systems, with immediate applicability in near-term quantum hardware.

Figures (9)

• Figure 1: Schematic of QRC architecture. Quantum reservoir comprises two subsystems: (i) input qubits (blue) for encoding input variables through Pauli $Y$-rotation $R_Y$, and (ii) hidden qubits (pink) storing historical information to predict future outcomes. This process begins at time step $t{-}d$ and continues until $t{-}1$, such that the total number of quantum layers is $d$. After the evolutions of $d$ sequential layers, quantum measurements along Pauli $X$-directions are implement on all the input and hidden qubits.
• Figure 2: The bifurcation diagram of the logistic map (a) and the Hénon map (b). The largest Lyapunov exponent (dark line) of the logistic map (c) and the henon map (d). The regions with a white background indicate LLE values less than 0, while the grey background highlights regions where the LLE is greater than 0. Additionally, the RMSE of predicting the bifurcation diagrams have shown as the blue lines in (c) and (d).
• Figure 3: The effect of number of layers and repetitions on QRC for the logistic and Hénon map for (a) and (b) predicting $x_t$ and (c) and (d) predicting $x_{t+1}$ for $r=3.75$ ($a = 1.35$). Darker hues indicate lower RMSE (better predictive accuracy). Lowest RMSE is for $2$ layers for logistic map ($1$ layer for Hénon map) and $n_{\textrm{rep}}{=}2$ for predicting $x_t$ and $n_{\textrm{rep}}{=}4$ for predicting $x_{t+1}$ in both cases.
• Figure 4: Actual (red) vs predicted (blue) bifurcation diagram of the logistic map with different closed-loop prediction horizons $k$, where for (a) $k = 3$, (b) $k = 5$, (c) $k = 7$ and (d) $k = 9$. Actual (red) vs predicted (blue) bifurcation diagram of the Hénon map with different closed-loop prediction horizons $k$, where for (e) $k = 3$, (f) $k = 5$, (g) $k = 7$, and (h) $k = 9$.
• Figure 5: (a) RMSE of predicting bifurcation diagram of the logistic map with different horizon steps $k = 3,5,7,9$. LLE (black line) and chaotic regimes (grey) are depicted for reference. (b) RMSE of prediction at different regimes $r=2.9$ (fixed point, blue), $r=3.5$ (4-periodic orbit, teal), $r=3.829$ (island of stability, magenta), $r=4.0$ (chaotic, red) with prediction horizon $k$ ranging from 1 to 10. (c) same as (a) but for Henon map. (d) same as (b) but for Henon map at different regimes $a = 1.005$ (4-periodic orbit, teal) , $a = 1.25$ (7-periodic orbit, blue), $a = 1.4$ (chaotic, magenta).