Linear regression is a widely used technique to fit linear models and finds widespread applications across different areas such as machine learning and statistics. In most real-world scenarios, however, linear regression problems are often ill-posed or the underlying model suffers from overfitting, leading to erroneous or trivial solutions. This is often dealt with by adding extra constraints, known as regularization. In this paper, we use the frameworks of block-encoding and quantum singular value transformation (QSVT) to design the first quantum algorithms for quantum least squares with general $\ell_2$-regularization. These include regularized versions of quantum ordinary least squares, quantum weighted least squares, and quantum generalized least squares. Our quantum algorithms substantially improve upon prior results on quantum ridge regression (polynomial improvement in the condition number and an exponential improvement in accuracy), which is a particular case of our result. To this end, we assume approximate block-encodings of the underlying matrices as input and use robust QSVT algorithms for various linear algebra operations. In particular, we develop a variable-time quantum algorithm for matrix inversion using QSVT, where we use quantum eigenvalue discrimination as a subroutine instead of gapped phase estimation. This ensures that substantially fewer ancilla qubits are required for this procedure than prior results. Owing to the generality of the block-encoding framework, our algorithms are applicable to a variety of input models and can also be seen as improved and generalized versions of prior results on standard (non-regularized) quantum least squares algorithms.

翻译：线性回归是一种广泛用于拟合线性模型的技术，在机器学习和统计学等多个领域均有广泛应用。然而，在大多数实际场景中，线性回归问题往往是不适定的，或潜在模型存在过拟合，从而导致错误或平凡解。通常通过添加额外约束（即正则化）来处理这一问题。本文利用块编码和量子奇异值变换（QSVT）框架，首次设计了针对一般$\ell_2$正则化的量子最小二乘问题的量子算法。这些算法包括正则化的量子普通最小二乘法、量子加权最小二乘法以及量子广义最小二乘法。我们的量子算法显著改进了先前关于量子岭回归的结果（条件数上实现多项式改进，精度上实现指数级改进），而量子岭回归正是我们结果的一个特例。为此，我们假设输入为底层矩阵的近似块编码，并针对各种线性代数运算采用鲁棒的QSVT算法。特别地，我们利用QSVT开发了一种可变时间的量子矩阵求逆算法，该算法使用量子特征值判别作为子程序，而非有间隙的相位估计。这确保了该过程所需的辅助量子比特数量远少于先前结果。得益于块编码框架的通用性，我们的算法适用于多种输入模型，也可被视为先前标准（非正则化）量子最小二乘算法结果的改进与推广版本。