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. 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.

翻译：量子极端学习机（QELMs）已成为量子机器学习领域一种极具前景的框架。其魅力源于量子基底（即量子储层）动态过程所引发的丰富特征映射，以及通过线性回归实现的高效测量后训练。本文通过将QELMs的预测结果分解为傅里叶级数来研究其表达能力。我们证明，可实现的傅里叶频率由数据编码方案决定，而傅里叶系数则同时依赖于储层和测量方式。值得注意的是，QELMs的表达能力从根本上受限于傅里叶频率的数量与可观测量的数量，而预测的复杂性则取决于储层。作为对可扩展性的警示，我们确定了四种可能导致可观测量随系统规模增大而呈指数级集中的因素（随机性、硬件噪声、纠缠以及全局测量），并展示了这会如何使QELMs沦为无用的输入无关预言机。我们的分析阐明了QELMs的潜力与根本局限性，并为系统探索适用于其他机器学习任务的量子储层系统奠定了基础。