Krylov Expressivity in Quantum Reservoir Computing and Quantum Extreme Learning

Saud Čindrak, Lina Jaurigue, Kathy Lüdge
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Abstract

Quantum machine learning utilizes the high-dimensional space of quantum systems, attracting significant research interest. This study employs Krylov complexity to analyze task performance in quantum machine learning. We calculate the spread complexity and effective dimension of the Krylov space, introducing the effective dimension as an easy-to-compute, measurable, and upper-bounded expressivity measure. Our analysis covers quantum reservoir computers and quantum extreme learning machines, showing that increasing effective dimension correlates with improved performance. We validate this with the Lorenz cross-prediction task, observing reduced error with higher effective dimensions. Lastly, we compare the spread complexity, the effective dimension, and the fidelity as expressivity measures and show that fidelity is not suitable, while spread complexity can qualitatively explain task performance. Only the effective dimension captures the phase space accurately and exhibits the same saturation as task performance for similar evolution times.
量子存储计算和量子极限学习中的克雷洛夫表达式
量子机器学习利用量子系统的高维空间,吸引了大量研究兴趣。本研究利用克雷洛夫复杂性分析量子机器学习中的任务性能。我们计算了克雷洛夫空间的扩散复杂度和有效维度,并引入有效维度作为一种易于计算、可测量和上界表达度量。我们的分析涵盖了量子储备计算机和量子极端学习机,表明有效维度的增加与性能的提高相关。我们用洛伦兹交叉预测任务验证了这一点,发现有效维度越高,错误越少。最后,我们比较了作为表达度量的扩展复杂度、有效维度和保真度,结果表明保真度并不合适,而扩展复杂度可以定性地解释任务性能。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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