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.

翻译：量子储层计算机（QRCs）已成为量子机器学习的一种有前景的方法，因为它们利用量子系统的自然动力学进行数据处理且训练简单。本文研究具有连续时间储层动力学的n量子比特量子极限学习机（QELMs）。QELMs是无记忆的QRCs，能够执行多种机器学习任务，包括图像分类和时间序列预测。我们应用泡利转移矩阵（PTM）形式体系，从理论上分析编码、储层动力学和测量操作（包括时间复用）对QELM性能的影响。该形式体系明确指出：编码决定了QELM可用的完整（非线性）特征集，而量子信道在所选测量算符探测这些特征之前对其进行线性变换。因此，优化QELM可转化为一个解码问题，即通过设计信道诱导的变换，使任务相关特征能够被回归器利用。PTM形式体系允许识别QELM的经典表示，从而指导其针对给定训练目标进行设计。作为一个具体应用，我们专注于学习非线性动力系统，并证明在此类轨迹上训练的QELM能够学习到底层流映射的代理近似。