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Theory and interpretability of Quantum Extreme Learning Machines: a Pauli-transfer matrix approach

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 initial-state encoding and continuous-time reservoir dynamics. 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 reveals the complete set of (nonlinear) features generated by the encoding, and shows how the subsequent quantum channels linearly transform these Pauli features before they are probed by the chosen measurement operators. Optimizing such 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, effectively reversing the information scrambling of a unitary. Operator spreading under unitary evolution determines decodability of Pauli features, which underlies the nonlinear processing capacity of the reservoir. When paired with certain observables, structured Hamiltonians can reduce model expressivity, as reflected in a low readout rank. We trace this effect to Hamiltonian symmetries and derive asymptotic rank estimates for symmetry-resolved observable families. The PTM formalism yields a nonlinear vector (auto-)regression model as an interpretable classical representation of a QELM. As a specific application, we focus on forecasting nonlinear dynamical systems and show that a QELM trained on such trajectories learns a surrogate-approximation to the underlying flow map.

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