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. Applying the Pauli transfer matrix (PTM) formalism reveals the encoding-generated features and their linear transformation by quantum channels, and thereby provides a nonlinear vector autoregression model as an interpretable classical representation of a QELM. We show that the nonlinear information processing capacity of a QELM factorizes into the ability of the encoding to represent a given target function and the decodability of the relevant Pauli features after the effective PTM. Comparing geometric and statistical decodability reveals features that are algebraically hidden yet recoverable through data correlations. We relate the decodability of Pauli features to the spreading of measurement operators under Heisenberg evolution (Krylov growth). Focusing on the unitary channel, we discuss how structured Hamiltonians, when paired with certain observables, can impair model performance, as reflected in a low rank of the readout and the effective PTMs. We trace this effect to Hamiltonian symmetries and derive asymptotic rank estimates for symmetry-resolved observable families. Our decodability analysis disentangles the contributions of encoding, reservoir dynamics, and measurement observables, and thus provides interpretable diagnostics that can inform the design of QELMs and reduce reliance on exhaustive hyperparameter search. As a specific application, we consider forecasting nonlinear dynamical systems and show that the readout of a trained QELM closely matches the expression of the underlying flow map at leading orders.
Fidelity is crucial for characterizing transformations of quantum states under various quantum channels, which can be served as a fundamental tool in resource theories. Firstly, we define an
α-
z-fidelity as a significant quantity in quantum information theory and give the properties of the fidelity with orders
α and
z. Secondly, by analyzing the
α-
z-fidelity under the evolution of different types of quantum channels (single orbit, all quantum channels, unitary quantum channels, and mixed unitary quantum channels), we propose a limit formula for the maximum and the minimum of the
α-
z-fidelity. In addition, we have extended the
α-
z-Rényi relative entropy, providing new insights into its relevance for resource quantification. Finally, we offer a geometric interpretation for measuring the distance between quantum states, contributing to the broader understanding of the operational and transformative power of dynamical quantum resources across various physical settings.
A geometric interpretation for the
A fidelity between two states of a qubit system is presented, which leads to an upper bound of the Bures fidelity. The metrics defined based on the
A fidelity are studied by the numerical method. An alternative generalization of the
A fidelity, which has the same geometric picture, to an
N-state quantum system is also discussed.