Robust quantum reservoir computers for forecasting chaotic dynamics: generalized synchronization and stability

We show that recurrent quantum reservoir computers (QRCs) and their recurrence-free architectures (RF-QRCs) are robust tools for learning and forecasting chaotic dynamics from time-series data. First, we formulate and interpret QRCs as coupled dynamical systems, where the reservoir acts as a response system driven by training data; in other words, QRCs are generalized-synchronization (GS) systems. Second, we show that QRCs can learn chaotic dynamics and their invariant properties, such as Lyapunov spectra, attractor dimensions and geometric properties such as the covariant Lyapunov vectors (CLVs). This analysis is enabled by deriving the Jacobian of the quantum reservoir update. Third, by leveraging tools from GS, we provide a method for designing robust QRCs. We propose the criterion <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> <mml:mi>S</mml:mi> <mml:mo>=</mml:mo> <mml:mi>E</mml:mi> <mml:mi>S</mml:mi> <mml:mi>P</mml:mi> </mml:math> : GS implies the echo state property (ESP) and vice versa. We analytically show that RF-QRCs, by design, fulfill <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> <mml:mi>S</mml:mi> <mml:mo>=</mml:mo> <mml:mi>E</mml:mi> <mml:mi>S</mml:mi> <mml:mi>P</mml:mi> </mml:math> . Finally, we analyze the effect of simulated noise. We find that dissipation from noise enhances the robustness of QRCs. Numerical verifications on systems of different dimensions support our conclusions. This work opens opportunities for designing robust quantum machines for chaotic time-series forecasting on near-term quantum hardware.