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

Abstract

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. QELMs are memoryless QRCs capable of various ML tasks, such as 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 this type of QELM, while the quantum channels linearly transform these 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. We show how operator spreading under unitary evolution determines feature decodability, which underlies the nonlinear processing capacity of the reservoir. The PTM formalism yields a nonlinear vector (auto-)regression model as the classical representation of a QELM. 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.

Reproductions