Topological data analysis (TDA) is a powerful technique for extracting complex and valuable shape-related summaries of high-dimensional data. However, the computational demands of classical algorithms for computing TDA are exorbitant, and quickly become impractical for high-order characteristics. Quantum computers offer the potential of achieving significant speedup for certain computational problems. Indeed, TDA has been purported to be one such problem, yet, quantum computing algorithms proposed for the problem, such as the original Quantum TDA (QTDA) formulation by Lloyd, Garnerone and Zanardi, require fault-tolerance qualifications that are currently unavailable. In this study, we present NISQ-TDA, a fully implemented end-to-end quantum machine learning algorithm needing only a short circuit-depth, that is applicable to high-dimensional classical data, and with provable asymptotic speedup for certain classes of problems. The algorithm neither suffers from the data-loading problem nor does it need to store the input data on the quantum computer explicitly. The algorithm was successfully executed on quantum computing devices, as well as on noisy quantum simulators, applied to small datasets. Preliminary empirical results suggest that the algorithm is robust to noise.

翻译：拓扑数据分析（TDA）是一种从高维数据中提取复杂且有价值的形状相关摘要的强大技术。然而，经典算法计算TDA的计算需求极高，并且很快在处理高阶特征时变得不切实际。量子计算机为某些计算问题实现显著加速提供了潜力。事实上，TDA曾被认为是此类问题之一，然而，为这一问题提出的量子计算算法（例如Lloyd、Garnerone和Zanardi提出的原始量子TDA（QTDA）公式）需要当前尚不具备的容错能力。在本研究中，我们提出了NISQ-TDA，一种完全实现的端到端量子机器学习算法，它仅需较短的电路深度，适用于高维经典数据，并且对于某些问题类别具有可证明的渐近加速。该算法既不受数据加载问题的影响，也无需将输入数据显式存储在量子计算机上。该算法已成功在量子计算设备以及噪声量子模拟器上执行，并应用于小型数据集。初步实证结果表明，该算法对噪声具有鲁棒性。