Quantum reservoir computers (QRCs) have emerged as a promising approach to quantum machine learning, since they utilize the natural dynamics of quantum systems for data processing and are simple to train. Here, we consider n-qubit quantum extreme learning machines (QELMs) with continuous-time reservoir dynamics. QELMs are memoryless QRCs capable of various ML tasks, including image classification and time series forecasting. We apply the Pauli transfer matrix (PTM) formalism to theoretically analyze the influence of encoding, reservoir dynamics, and measurement operations, including temporal multiplexing, on the QELM performance. This formalism makes explicit that the encoding determines the complete set of (nonlinear) features available to the QELM, while the quantum channels linearly transform these features before they are probed by the chosen measurement operators. Optimizing a QELM can therefore be cast as a decoding problem in which one shapes the channel-induced transformations such that task-relevant features become available to the regressor. The PTM formalism allows one to identify the classical representation of a QELM and thereby guide its design towards a given training objective. As a specific application, we focus on learning nonlinear dynamical systems and show that a QELM trained on such trajectories learns a surrogate-approximation to the underlying flow map.
Quantum reservoir computers (QRCs) have emerged as a promising approach to quantum machine learning, since they utilize the natural dynamics of quantum systems for data processing and are simple to train. Here, we consider n-qubit quantum extreme learning machines (QELMs) with continuous-time reservoir dynamics. QELMs are memoryless QRCs capable of various ML tasks, including image classification and time series forecasting. We apply the Pauli transfer matrix (PTM) formalism to theoretically analyze the influence of encoding, reservoir dynamics, and measurement operations, including temporal multiplexing, on the QELM performance. This formalism makes explicit that the encoding determines the complete set of (nonlinear) features available to the QELM, while the quantum channels linearly transform these features before they are probed by the chosen measurement operators. Optimizing a QELM can therefore be cast as a decoding problem in which one shapes the channel-induced transformations such that task-relevant features become available to the regressor. The PTM formalism allows one to identify the classical representation of a QELM and thereby guide its design towards a given training objective. As a specific application, we focus on learning nonlinear dynamical systems and show that a QELM trained on such trajectories learns a surrogate-approximation to the underlying flow map.
Quantum Machine Learning (QML) has emerged as a promising framework for exploring how quantum dynamics may enhance data processing tasks. Here we investigate Quantum Extreme Learning Machines (QELMs), a quantum analogue of classical Extreme Learning Machines in which training is restricted to the output layer. Our architecture combines dimensionality reduction (via PCA or Autoencoders), quantum state encoding, evolution under an XX Hamiltonian, and projective measurement to produce features for a classical single-layer classifier. By analyzing the classification accuracy as a function of evolution time, we observe a sharp transition between low- and high-accuracy regimes, followed by saturation. Remarkably, the saturated performance is comparable to that obtained using Haar-random unitaries that generate maximally complex dynamics, even though the XX model is integrable and local. Our results indicate that this increase in performance correlates with the onset of entanglement, which improves the embedding of classical data in Hilbert space and leads to more separable clusters in measurement probability space. Thus, moderate entanglement can contribute positively to the structure of the data representation, improving learnability without necessarily implying quantum computational advantage. For the image-classification tasks studied here, namely MNIST, Fashion-MNIST, and CIFAR-10, the relevant evolution time is consistent with information exchange over short distances and, within the explored system sizes, does not show evidence of scaling with the full system size. This suggests that QELM performance in this regime relies only on limited entanglement and remains compatible with efficient classical simulation. Our results clarify how local quantum dynamics and moderate quantum correlations are already sufficient to generate useful feature representations for learning.
Reservoir computing leverages rich, non-linear dynamics to process temporal data. Quantum variants promise enhanced expressivity from high-dimensional Hilbert spaces, yet their practical applicability is hindered by hardware noise and concentration effects that can erase input-output distinguishability at large system sizes. In this work, we present and experimentally demonstrate a Quantum Extreme Learning Machine (QELM) tailored to state-of-the-art superconducting platforms, employing up to 124 qubits and circuits with more than 5,000 two-qubit gates on IBM Quantum computers. We introduce a practical multi-objective hyperparameter tuning strategy that jointly monitors observable variability, capacity, and task performance to identify noise-robust operating points. In addition, we develop a local eigentask analysis that enables computationally efficient feature selection and effective information retrieval. We report evidence of a regime of optimality that is identifiable at small scales and transferable across tasks and larger systems, and we achieve performances competitive with leading classical baselines on representative benchmarks for time-series forecasting and satellite image classification. Together, our results establish a viable and robust framework for large-scale, pre-fault-tolerant quantum machine learning and provide a foundation for extending reservoir-based methods to more expressive architectures and real-world scenarios.
Abstract Quantum extreme learning machines (QELMs) have emerged as a promising framework for quantum machine learning. Their appeal lies in the rich feature map induced by the dynamics of a quantum substrate—the quantum reservoir—and the efficient post-measurement training via linear regression. Here, we study the expressivity of QELMs by decomposing the prediction of QELMs into a Fourier series. We show that the achievable Fourier frequencies are determined by the data encoding scheme, while Fourier coefficients depend on both the reservoir and the measurement. Notably, the expressivity of QELMs is fundamentally limited by the number of Fourier frequencies and the number of observables, while the complexity of the prediction hinges on the reservoir. As a cautionary note on scalability, we identify four sources that can lead to the exponential concentration of the observables as the system size grows (randomness, hardware noise, entanglement, and global measurements) and show how this can turn QELMs into useless input-agnostic oracles. In particular, our result on the reservoir-induced concentration strongly indicates that quantum reservoirs drawn from a highly random ensemble make QELM models unscalable. Our analysis elucidates the potential and fundamental limitations of QELMs and lays the groundwork for systematically exploring quantum reservoir systems for other machine learning tasks.
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.
Quantum reservoir computers (QRCs) retain information from earlier inputs, whereas reset-window quantum extreme learning machines (QELMs) start afresh and receive only a recent input window. We establish when this window can replace the reservoir's memory. Under our stated assumptions and with exact expectation values, sufficiently expressive QELMs can reproduce reservoir outputs arbitrarily accurately as their windows grow precisely when the influence of the omitted past becomes uniformly negligible. However, longer windows do not always solve the problem. We construct a simple two-qubit reservoir that performs temporal recall exactly, while every fixed-window predictor faces the same positive worst-case error over unbounded delays, regardless of model capacity. Analytical benchmarks and simulations distinguish missing history from limited features and finite measurements. The central conclusion is simple: better representations can improve how available information is used, but cannot recover a past the model never receives.
Quantum extreme learning machines (QELMs) leverage untrained quantum dynamics to efficiently process information encoded in input quantum states, avoiding the high computational cost of training more complicated nonlinear models. On the other hand, quantum information scrambling (QIS) quantifies how the spread of quantum information into correlations makes it irretrievable from local measurements. Here, we explore the tight relation between QIS and the predictive power of QELMs. In particular, we show efficient state estimation is possible even beyond the scrambling time, for many different types of dynamics — in fact, we show that in all the cases we studied, the reconstruction efficiency at long interaction times matches the optimal one offered by random global unitary dynamics. These results offer promising venues for robust experimental QELM-based state estimation protocols, as well as providing novel insights into the nature of QIS from a state estimation perspective.