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.
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.
In chaotic dynamical systems, extreme events manifest in time series as unpredictable large-amplitude peaks. Although deterministic, extreme events appear seemingly randomly, which makes their forecasting difficult. By learning the dynamics from observables (data), reservoir computers can time accurately predict extreme events and chaotic dynamics, but they may require many degrees of freedom (large reservoirs). In this paper, by exploiting quantum-computer ansätze and entanglement, we design reservoir computers with compact reservoirs and accurate prediction capabilities. First, we propose the recurrence-free quantum reservoir computer (RF-QRC) architecture. By developing quantum feature maps and removing recurrent connections, the RF-QRC has quantum circuits with smaller depths. This allows the RF-QRC to scale well with higher-dimensional chaotic systems, which makes it suitable for hardware implementation. Second, we forecast the temporal chaotic dynamics and their long-term statistics of low- and higher-dimensional dynamical systems. We find that RF-QRC requires smaller reservoirs than classical reservoir computers for higher-dimensional systems and the same predictability. Third, we apply the RF-QRC to the time prediction of extreme events in a model of a turbulent shear flow with turbulent bursts. We find that the RF-QRC has longer predictability than the classical reservoir computer for extreme events forecasting. The results and analyses indicate that quantum-computer ansätze offers nonlinear expressivity and computational scalability, which are useful for forecasting chaotic dynamics and extreme events. This work opens new opportunities for using quantum machine learning on near-term quantum computers. Published by the American Physical Society 2024
Abstract In the current Noisy Intermediate Scale Quantum (NISQ) era, the presence of noise deteriorates the performance of quantum computing algorithms. Quantum reservoir computing (QRC) is a type of quantum machine learning algorithm, which, however, can benefit from different types of tuned noise. In this paper, we analyze how finite sampling noise affects the chaotic time series prediction of the gate-based QRC and recurrence-free quantum reservoir computing (RF-QRC) models. First, we examine RF-QRC and show that, even without a recurrent loop, it contains temporal information about previous reservoir states using leaky integrated neurons. This makes RF-QRC different from quantum extreme learning machines (QELM). Second, we show that finite sampling noise degrades the prediction capabilities of both QRC and RF-QRC while affecting QRC more due to the propagation of noise. Third, we optimize the training of the finite-sampled quantum reservoir computing framework using two methods: (a) singular value decomposition (SVD) applied to the data matrix containing noisy reservoir activation states and (b) data-filtering techniques to remove the high frequencies from the noisy reservoir activation states. We show that denoising reservoir activation states improves the signal-to-noise ratios with smaller training loss. Finally, we demonstrate that the training and denoising of the noisy reservoir activation signals in RF-QRC are highly parallelizable on multiple quantum processing units (QPUs) as compared to the QRC architecture with recurrent connections. The analyses are numerically showcased on prototypical chaotic dynamical systems with relevance to turbulence. This work opens opportunities for using quantum reservoir computing with finite samples for time series forecasting on near-term quantum hardware.
Forecasting dynamical systems is of importance to numerous real-world applications. Typically, these are obtained using first-principles-based models such as through the use of differential equations. When these equations are unknown, non-intrusive techniques must be utilized to build predictive models from data alone. Machine learning (ML) methods have recently been used for such tasks with the added advantage of significant reductions in time-to-solution for predictions during deployment. However, many state-of-the-art ML-based methods for forecasting rely on neural networks, which are expensive to train and require large amounts of memory. In this work, we propose a quantum mechanics inspired ML strategy for forecasting nonlinear dynamical systems that results in reduced training time and memory requirements. This approach, denoted the quantum reservoir computing technique (QRC), is a hybrid quantum-classical framework employing an ensemble of interconnected small quantum systems via classical linear feedback connections. By mapping the dynamical state to a suitable quantum representation amenable to unitary operations, QRC is able to predict complex nonlinear dynamical systems in a stable and accurate manner. We demonstrate the efficacy of this framework through benchmark forecasts of the NOAA Optimal Interpolation Sea Surface Temperature dataset and compare the performance of QRC to other ML methods.
Quantum reservoir computing (QRC) is an emerging paradigm for harnessing the natural dynamics of quantum systems as computational resources that can be used for temporal machine learning tasks. In the current setup, QRC is difficult to deal with high-dimensional data and has a major drawback of scalability in physical implementations. We propose higher-order QRC, a hybrid quantum-classical framework consisting of multiple but small quantum systems that are mutually communicated via classical connections like linear feedback. By utilizing the advantages of both classical and quantum techniques, our framework enables an efficient implementation to boost the scalability and performance of QRC. Furthermore, higher-order settings allow us to implement a FORCE learning or an innate training scheme, which provides flexibility and high operability to harness high-dimensional quantum dynamics and significantly extends the application domain of QRC. We demonstrate the effectiveness of our framework in emulating large-scale nonlinear dynamical systems, including complex spatiotemporal chaos, which outperforms many of the existing machine learning techniques in certain situations.
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 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.
[eng] Quantum technologies are rapidly evolving toward regimes where controllable quantum dynamics can be exploited for information processing, simulation, and machine learning. Within this landscape, quantum reservoir computing (QRC) has emerged as an appealing paradigm: it harnesses the high-dimensional dynamics of quantum systems to perform temporal tasks with minimal training overhead, making it particularly well suited for current noisy intermediate-scale quantum devices. In parallel, modern perspectives on quantum many-body systems reveal that tools such as complex-network theory and spectral methods can uncover structures in quantum states and dynamics that remain hidden to conventional approaches. This thesis brings these threads together. It explores how quantum statistics, many-body correlations, chaoticity, and metastability shape information processing in QRC. Furthermore, we have also demonstrated how network-theoretic methods can reveal new features of strongly correlated phases. This thesis contributes to the emerging field of quantum reservoir computing by examining how different physical models shape the learning capabilities of quantum reservoirs. Each article investigates a distinct class of systems: simple hopping networks governed by particle statistics, interacting atomic lattices described by the Bose–Hubbard model, driven-dissipative oscillators exhibiting metastability. Through these diverse settings, a central goal is to identify which physical ingredients enhance the performance of QRC. To address these questions, we combine analytical insights with a broad range of numerical techniques, including exact diagonalization, tensor-network simulations, and open-system dynamics. Together, these approaches provide a unified framework for understanding how quantum dynamics can be exploited for computation. The main findings of the thesis on QRC and complex networks can be summarized as follows. (i) Particle statistics fundamentally shape computational power: fermions exhibit superior information spreading due to their algebra, while bosons can outperform all other substrates when input encoding is optimized. (ii) Metastability constitutes a computational resource: long-lived metastable states enhance memory capacity and allow a fast-injection protocol. (iii) In atomic lattices described by the Bose–Hubbard model, QRC performance is governed by the interplay between interactions and tunneling, with optimal regimes appearing both in the weakly interacting limit and near the onset of quantum chaos; importantly, disorder is not required to achieve high performance. (iv) Complex-network analysis reveals hidden features of the Kitaev chain: network observables sharply detect the topological transition in the Kitaev chain and uncover a previously unnoticed fully connected, perfectly clustered regime linked to ground-state factorization observed in the Ising spin chain.