{"title":"Krylov Expressivity in Quantum Reservoir Computing and Quantum Extreme Learning","authors":"Saud Čindrak, Lina Jaurigue, Kathy Lüdge","doi":"arxiv-2409.12079","DOIUrl":null,"url":null,"abstract":"Quantum machine learning utilizes the high-dimensional space of quantum\nsystems, attracting significant research interest. This study employs Krylov\ncomplexity to analyze task performance in quantum machine learning. We\ncalculate the spread complexity and effective dimension of the Krylov space,\nintroducing the effective dimension as an easy-to-compute, measurable, and\nupper-bounded expressivity measure. Our analysis covers quantum reservoir\ncomputers and quantum extreme learning machines, showing that increasing\neffective dimension correlates with improved performance. We validate this with\nthe Lorenz cross-prediction task, observing reduced error with higher effective\ndimensions. Lastly, we compare the spread complexity, the effective dimension,\nand the fidelity as expressivity measures and show that fidelity is not\nsuitable, while spread complexity can qualitatively explain task performance.\nOnly the effective dimension captures the phase space accurately and exhibits\nthe same saturation as task performance for similar evolution times.","PeriodicalId":501226,"journal":{"name":"arXiv - PHYS - Quantum Physics","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - PHYS - Quantum Physics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.12079","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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.