Fast Quantum Algorithms for Least Squares Regression and Statistic Leverage Scores

Abstract

Least squares regression is the simplest and most widely used technique for solving overdetermined systems of linear equations \(Ax = b\), where \(A\in \mathbb R^{n\times p}\) has full column rank and \(b\in \mathbb R^n\). Though there is a well known unique solution \(x^*\in \mathbb R^p\) to minimize the squared error \(\Vert Ax - b\Vert _2^2\), the best known classical algorithm to find \(x^*\) takes time \(\Omega (n)\), even for sparse and well-conditioned matrices \(A\), a fairly large class of input instances commonly seen in practice. In this paper, we design an efficient quantum algorithm to generate a quantum state proportional to \(| x^* \rangle \). The algorithm takes only \(O(\log n)\) time for sparse and well-conditioned \(A\). When the condition number of \(A\) is large, a canonical solution is to use regularization. We give efficient quantum algorithms for two regularized regression problems, including ridge regression and \(\delta \)-truncated SVD, with similar costs and solution approximation.