量子辅助的希尔伯特空间高斯过程回归
Quantum-Assisted Hilbert-Space Gaussian Process Regression
AI总结：
针对高斯过程回归随数据集增长计算复杂度难处理的问题，提出结合经典基函数展开与多种量子计算技术的希尔伯特空间近似量子算法，实现多项式级复杂度降低。
摘要：
高斯过程是机器学习中常用作函数先验的概率模型。凭借其概率特性，它们可用于捕捉关于噪声统计特性、函数平滑度以及训练数据不确定性的先验信息。然而，随着数据集规模的增长，其计算复杂度会迅速变得难以处理。我们提出了一种基于希尔伯特空间近似的量子算法，用于高斯过程回归以克服这一限制。我们的方法结合了经典基函数展开与量子主成分分析、条件旋转以及Hadamard检验和Swap检验等量子计算技术。量子主成分分析用于估计特征值，而条件旋转以及Hadamard检验和Swap检验则用于评估高斯过程的后验均值和方差。与经典方法相比，我们的方法实现了多项式级的计算复杂度降低。
英文摘要：
Gaussian processes are probabilistic models that are commonly used as functional priors in machine learning. Due to their probabilistic nature, they can be used to capture the prior information on the statistics of noise, smoothness of the functions, and training data uncertainty. However, their computational complexity quickly becomes intractable as the size of the data set grows. We propose a Hilbert space approximation-based quantum algorithm for Gaussian process regression to overcome this limitation. Our method consists of a combination of classical basis function expansion with quantum computing techniques of quantum principal component analysis, conditional rotations, and Hadamard and Swap tests. The quantum principal component analysis is used to estimate the eigenvalues while the conditional rotations and the Hadamard and Swap tests are employed to evaluate the posterior mean and variance of the Gaussian process. Our method provides polynomial computational complexity reduction over the classical method.
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