Topological data analysis (TDA) has emerged as a powerful tool for extracting meaningful insights from complex data. TDA enhances the analysis of objects by embedding them into a simplicial complex and extracting useful global properties such as the Betti numbers, i.e. the number of multidimensional holes, which can be used to define kernel methods that are easily integrated with existing machine-learning algorithms. These kernel methods have found broad applications, as they rely on powerful mathematical frameworks which provide theoretical guarantees on their performance. However, the computation of higher-dimensional Betti numbers can be prohibitively expensive on classical hardware, while quantum algorithms can approximate them in polynomial time in the instance size. In this work, we propose a quantum approach to defining topological kernels, which is based on constructing Betti curves, i.e. topological fingerprint of filtrations with increasing order. We exhibit a working prototype of our approach implemented on a noiseless simulator and show its robustness by means of some empirical results suggesting that topological approaches may offer an advantage in quantum machine learning.

翻译：拓扑数据分析（TDA）已成为从复杂数据中提取有意义信息的有力工具。TDA通过将数据对象嵌入单纯复形并提取诸如贝蒂数（即多维空洞的数量）等有用的全局属性，从而增强数据分析能力。这些属性可用于定义与现有机器学习算法轻松集成的核方法。由于依赖提供理论性能保障的强大数学框架，这些核方法已获得广泛应用。然而，在经典硬件上计算高阶贝蒂数的成本可能极高，而量子算法能在实例规模的多项式时间内逼近这些数值。在本工作中，我们提出一种基于量子方法定义拓扑核的方案，该方案通过构造贝蒂曲线（即随阶数递增的滤子拓扑指纹）来实现。我们展示了该方案在无噪声模拟器上的工作原型，并通过实证结果证明其鲁棒性，表明拓扑方法可能在量子机器学习中具有优势。