通过量子拓扑分析的实现理解编码数据的映射
Understanding the Mapping of Encode Data Through An Implementation of Quantum Topological Analysis
AI总结：
提出一种混合量子拓扑分析方法，通过持续 Betti 数与持续同调图比较角度、振幅和 IQP 编码在 Hilbert 空间中的拓扑差异，强调编码选择对下游任务的关键影响。
摘要：
量子机器学习的一个潜在优势源于利用量子电路将经典数据编码到高维复 Hilbert 空间的能力。近期研究表明，在表示经典数据时，并非所有编码方法都相同，因为某些参数化电路结构比其他结构具有更强的表达能力。在本研究中，我们表明，通过考察嵌入复 Hilbert 空间中的数据拓扑，可以直观呈现不同编码技术之间的差异。用于可视化的技术是一种基于量子的混合拓扑分析，它利用边界算子的简单对角化来计算持续 Betti 数和持续同调图。为了在 NISQ 框架内增强 Betti 数的计算，我们提出了一种简单的混合算法。通过一个合成数据集的启发性示例，以及角度编码、振幅编码和 IQP 编码方法，我们揭示了各编码方法之间以及其与原始数据之间的拓扑差异。因此，我们的结果表明，在不同的量子机器学习模型中需要谨慎考虑编码方法，因为它会强烈影响聚类或分类等下游分析。
英文摘要：
A potential advantage of quantum machine learning stems from the ability of encoding classical data into high dimensional complex Hilbert space using quantum circuits. Recent studies exhibit that not all encoding methods are the same when representing classical data since certain parameterized circuit structures are more expressive than the others. In this study, we show the difference in encoding techniques can be visualized by investigating the topology of the data embedded in complex Hilbert space. The technique for visualization is a hybrid quantum based topological analysis which uses simple diagonalization of the boundary operators to compute the persistent Betti numbers and the persistent homology graph. To augment the computation of Betti numbers within a NISQ framework, we suggest a simple hybrid algorithm. Through a illuminating example of a synthetic data set and the methods of angle encoding, amplitude encoding, and IQP encoding, we reveal topological differences with the encoding methods, as well as the original data. Consequently, our results suggest the encoding method needs to be considered carefully within different quantum machine learning models since it can strongly affect downstream analysis like clustering or classification.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
