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
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
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.
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.
The prediction of complex dynamics remains an open problem across many domains of physics, where nonlinearities and multiscale interactions severely limit the reliability of conventional forecasting methods. Quantum Reservoir Computing has emerged as a promising paradigm for information processing by exploiting the nonlinear dynamical evolution of quantum systems to generate expressive feature representations of input signals. Here, we introduce a hybrid quantum-classical reservoir architecture capable of handling multivariate time series through quantum evolution combined with classical memory enhancement. Our model employs a five-qubit transverse-field Ising Hamiltonian with input-modulated dynamics and temporal multiplexing. We apply this framework to two paradigmatic models of chaotic behavior in fluid dynamics, where multiscale dynamics and nonlinearities play a dominant role: a low-dimensional truncation of the two-dimensional Navier-Stokes equations and the Lorenz-63 system. By systematically scanning the quantum system's parameter space, we identify regions that maximize forecasting performance. The observed robustness and reliable performances for both dynamical systems suggest that this hybrid quantum approach offers a flexible platform for modeling complex nonlinear time series.
We investigate quantum reservoir computing (QRC) using a hybrid qubit-boson system described by the Jaynes-Cummings (JC) Hamiltonian and its dispersive limit (DJC). These models provide high-dimensional Hilbert spaces and intrinsic nonlinear dynamics, making them powerful substrates for temporal information processing. We systematically benchmark both reservoirs through linear and nonlinear memory tasks, demonstrating that they exhibit an unusual superior nonlinear over linear memory capacity. We further test their predictive performance on the Mackey-Glass time series, a widely used benchmark for chaotic dynamics, and show comparable forecasting ability. We also investigate how memory and prediction accuracy vary with reservoir parameters and show the role of higher-order bosonic observables and time multiplexing in enhancing expressivity, even in minimal spin-boson configurations. Our results establish JC- and DJC-based reservoirs as versatile platforms for time-series processing and as elementary units that overcome the setting of equivalent qubit pairs and offer pathways toward tunable, high-performance quantum machine learning architectures.
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.
Reservoir computing (RC) is an effective method for predicting chaotic systems by using a high-dimensional dynamic reservoir with fixed internal weights, while keeping the learning phase linear, which simplifies training and reduces computational complexity compared to fully trained recurrent neural networks (RNNs). Quantum reservoir computing (QRC) uses the exponential growth of Hilbert spaces in quantum systems, allowing for greater information processing, memory capacity, and computational power. However, the original QRC proposal requires coherent injection of inputs multiple times, complicating practical implementation. We present a hybrid quantum-classical approach that implements memory through classical post-processing of quantum measurements. This approach avoids the need for multiple coherent input injections and is evaluated on benchmark tasks, including the chaotic Mackey-Glass time series prediction. We tested our model on two physical platforms: a fully connected Ising model and a Rydberg atom array. The optimized model demonstrates promising predictive capabilities, achieving a higher number of steps compared to previously reported approaches.