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.
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.
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
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.
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.
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.
The emerging technology of Quantum Reservoir Computing (QRC) stands out in the noisy-intermediate scale quantum era for its efficiency, adaptability, and compatibility with current quantum hardware. By harnessing quantum dynamics for feature extraction, QRC offers a promising alternative for complex forecasting tasks — such as predicting agri-commodity price fluctuations — central to addressing food price crises and supply chain sustainability. In this work, we systematically investigate the design of quantum reservoirs and identify circuit complexity, quantified via the majorization criterion, as a practical indicator of forecasting performance. Unlike alternative complexity measures that require full state tomography or time-reversed dynamics, majorization can be evaluated from measurement statistics alone, making it particularly suited for near-term quantum devices. We demonstrate that quantum reservoirs built from a universal gate family composed of Hadamard, controlled-NOT, and T gates achieve strong predictive accuracy while requiring significantly fewer gates than the widely used transverse-field Ising model. Our results show that quantum reservoirs outperform deep learning and statistical baselines, and achieve comparable, slightly superior, performance to classical reservoir computing—an improvement consistent with the compact model structure of QRC, which, with significantly fewer trainable parameters, is better suited to prevent overfitting in data-scarce regimes. We further explore robustness under realistic noise conditions, showing that certain types of quantum noise, such as amplitude damping, can even enhance performance at low error rates. These findings suggest that QRC is a promising tool for time series prediction in data-limited, high-impact domains.
Quantum Reservoir Computing (QRC) processes temporal data by retaining a memory of past inputs in the recurrent state of a quantum system, whereas a Quantum Extreme-Learning Machine (QELM) discards that memory, resetting the system at every step so that only the most recent input shapes the response. The two are usually treated as separate computational paradigms. We show that they are the two limits of a single architecture, connected by the input-encoding length, that is, the number of qubits overwritten with fresh data at each step. When a single qubit is re-encoded the system operates as a standard QRC, when the whole register is re-encoded it operates as a QELM, and intermediate lengths interpolate between them. The overwritten qubits hold the recent past in an explicit register, while the remaining qubits are never reset and carry older inputs forward in their evolving quantum state, so the encoding length redistributes memory between explicit and recurrent storage at fixed system size. Tuning the reservoir Hamiltonian and the evolution time with Bayesian optimization at each encoding length, we find that recurrent quantum memory is essential when a task must reach far into the past, and dispensable when the relevant history is short, where the memoryless reset limit already suffices. For every task the best reservoirs operate at the edge of chaos, where they perform as well as a densely connected reservoir with random all-to-all couplings of the same size, indicating that what temporal processing requires is the dynamical regime rather than the connectivity.