Forecasting of chaotic dynamics and extreme events with quantum reservoir computing and linear embeddings
File(s)
Author(s)
Ahmed, Osama
Type
Thesis or dissertation
Abstract
Forecasting chaotic dynamical systems and extreme events remains a central challenge in computational science, with applications spanning weather prediction, turbulence, and nonlinear physics. Classical methods suffer from the curse of dimensionality, while data-driven approaches provide fast predictive capabilities but often lack scalability and physical interpretability. This thesis develops quantum computational approaches for modelling, forecasting, and analysing nonlinear dynamical systems to provide scalable and physically interpretable frameworks for chaotic dynamics and extreme events.
The first part of this thesis develops quantum reservoir computing (QRC) methods for forecasting high-dimensional chaotic dynamics and extreme events. Using the theory of generalised synchronisation (GS), we establish design principles for stable and learnable quantum reservoir computers and introduce recurrence-free quantum reservoir computers (RF-QRCs) as a scalable alternative, demonstrating that RF-QRCs outperform classical reservoir computers for high-dimensional forecasting. We establish practical criteria, generalised synchronisation = echo state property (GS=ESP), and derive analytical Jacobian formulations for stability analysis, investigate finite-sampling noise with denoising strategies, and demonstrate parallel implementations on quantum hardware. Depolarising and amplitude-damping channels are further analysed through GS theory to establish the noise--memory trade-off in quantum reservoir computing.
The second part of this thesis develops quantum-compatible approaches for solving nonlinear differential equations governing chaotic dynamics through linear operator embeddings based on the Fokker--Planck and Koopman--von Neumann formulations. We analyse higher-order discretisation schemes to study the effects of numerical diffusion and stability of probability-density evolution, establishing bounds linking numerical diffusion, grid resolution, and Lyapunov instability. We implement unitary block-encoding quantum algorithms for discretised Fokker--Planck dynamics, demonstrating how nonlinear chaotic evolution can be reformulated into sparse linear operator evolutions for near-term quantum computation.
Together, the complementary approaches developed in this thesis establish quantum computational frameworks for forecasting, analysing, and simulating nonlinear chaotic systems, while providing pathways toward practical quantum advantage in scientific computing.
The first part of this thesis develops quantum reservoir computing (QRC) methods for forecasting high-dimensional chaotic dynamics and extreme events. Using the theory of generalised synchronisation (GS), we establish design principles for stable and learnable quantum reservoir computers and introduce recurrence-free quantum reservoir computers (RF-QRCs) as a scalable alternative, demonstrating that RF-QRCs outperform classical reservoir computers for high-dimensional forecasting. We establish practical criteria, generalised synchronisation = echo state property (GS=ESP), and derive analytical Jacobian formulations for stability analysis, investigate finite-sampling noise with denoising strategies, and demonstrate parallel implementations on quantum hardware. Depolarising and amplitude-damping channels are further analysed through GS theory to establish the noise--memory trade-off in quantum reservoir computing.
The second part of this thesis develops quantum-compatible approaches for solving nonlinear differential equations governing chaotic dynamics through linear operator embeddings based on the Fokker--Planck and Koopman--von Neumann formulations. We analyse higher-order discretisation schemes to study the effects of numerical diffusion and stability of probability-density evolution, establishing bounds linking numerical diffusion, grid resolution, and Lyapunov instability. We implement unitary block-encoding quantum algorithms for discretised Fokker--Planck dynamics, demonstrating how nonlinear chaotic evolution can be reformulated into sparse linear operator evolutions for near-term quantum computation.
Together, the complementary approaches developed in this thesis establish quantum computational frameworks for forecasting, analysing, and simulating nonlinear chaotic systems, while providing pathways toward practical quantum advantage in scientific computing.
Version
Open Access
Date Issued
2026-05-26
Date Awarded
2026-09-01
Copyright Statement
Attribution-NonCommercial 4.0 International Licence (CC BY-NC)
License URL
Advisor
Magri, Luca
Publisher Department
Department of Aeronautics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)