Hybrid Quantum Deep Operational Echo State Operator for Chaotic Systems Prediction

Modeling nonlinear dynamical systems is central to physics, yet learning 

generalizable surrogates that remain stable across parameters and initial 

conditions remains challenging. Neural operator architectures such as 

DeepONets can approximate mappings between function spaces, but they 

struggle to encode temporal dependencies efficiently, especially in chaotic 

regimes where memory of the recent past determines future evolution.

To address this, we combine the dynamical memory of a quantum reservoir

with the operator-learning capability of a DeepONet. Our hybrid

architecture couples a Quantum Echo State Network (QESN), which transforms a short history of system states into a compact latent representation of the system’s phase, with a 

DeepONet that maps this latent state and the governing parameters 

to the continuous-time evolution of the system. This design enables a single 

network to learn a global operator for entire families of dynamical systems.

Here we show that both the classical and quantum ESN–DeepONet 

variants accurately predict short-term trajectories and reproduce the 

long-term statistical behavior of the Lorenz system across a wide range of 

parameters, while the quantum implementation achieves comparable accuracy 

with exponentially fewer internal resources. These results demonstrate that 

coupling quantum or classical recurrent dynamics with operator networks 

yields efficient, generalizable surrogates for chaotic and multiscale processes, paving the way toward quantum-enhanced operator learning and 

physics-informed digital twins of complex systems.