Dimensionality reduction (DR) techniques play an extremely critical role in the data mining and pattern recognition field. However, most DR approaches involve large-scale matrix computations, which cause too high running complexity to implement in the big data scenario efficiently. The recent developments in quantum information processing provide a novel path to alleviate this problem, where a potential quantum acceleration can be obtained comparing with the classical counterpart. Nevertheless, existing proposals for quantum DR methods faced the common dilemma of the nonlinear generalization owing to the intrinsic linear limitation of quantum computation. In this paper, an architecture to simulate the arbitrary nonlinear kernels on a universal quantum computer is illustrated and further propose the quantum kernel principal component analysis (QKPCA) algorithm. The key idea is employing the truncated Taylor expansion to approximate the arbitrary nonlinear kernel within the fixed error and then constructing the corresponding Hamiltonian simulation for the quantum phase estimation algorithm. It is demonstrated theoretically that the QKPCA is qualified for the nonlinear DR task while the exponential speedup is also maintained. In addition, this research has the potential ability to develop other quantum DR approaches and existing linear quantum machine learning models.

降维（DR）技术在数据挖掘和模式识别领域中发挥着至关重要的作用。但是，大多数灾难恢复方法都涉及大规模矩阵计算，这会导致运行复杂性过高而无法在大数据场景中有效实施。量子信息处理的最新发展为缓解这一问题提供了一条新颖的途径，与传统方法相比，该方法可以获得潜在的量子加速度。尽管如此，由于量子计算的内在线性限制，现有的关于量子DR方法的建议面临着非线性泛化的共同难题。在本文中，阐述了一种在通用量子计算机上模拟任意非线性核的体系结构，并进一步提出了量子核主成分分析（QKPCA）算法。关键思想是采用截断的泰勒展开来逼近固定误差内的任意非线性核，然后为量子相位估计算法构造相应的汉密尔顿模拟。从理论上证明，QKPCA可以胜任非线性DR任务，同时还能保持指数加速。此外，这项研究具有开发其他量子DR方法和现有线性量子机器学习模型的潜在能力。关键思想是采用截断的泰勒展开来逼近固定误差内的任意非线性核，然后为量子相位估计算法构建相应的汉密尔顿模拟。从理论上证明，QKPCA可以胜任非线性DR任务，同时还能保持指数加速。此外，这项研究具有开发其他量子DR方法和现有线性量子机器学习模型的潜在能力。关键思想是采用截断的泰勒展开来逼近固定误差内的任意非线性核，然后为量子相位估计算法构建相应的汉密尔顿模拟。从理论上证明，QKPCA可以胜任非线性DR任务，同时还能保持指数加速。此外，这项研究具有开发其他量子DR方法和现有线性量子机器学习模型的潜在能力。