Accelerating Regression Tasks with Quantum Algorithms

Regression is a cornerstone of statistics and machine learning, with applications spanning science, engineering, and economics. While quantum algorithms for regression have attracted considerable attention, most existing work has focused on linear regression, leaving many more complex yet practically important variants unexplored. In this work, we present a unified quantum framework for accelerating a broad class of regression tasks---including linear and multiple regression, Lasso, Ridge, Huber, $\ell_p$-, and $\delta_p$-type regressions---achieving up to a quadratic improvement in the number of samples $m$ over the best classical algorithms. This speedup is achieved by a non-trivial quantization of the recent classical breakthrough of Jambulapati et al. (2024), where we construct a full quantum pipeline that strategically employs quantum leverage score approximation to initialize and refine Multiscale Leverage Score Overestimates, enabling efficient importance sampling via the preparation of multiple state copies. For problems of dimension $n$, sparsity $r < n$, and error parameter $\epsilon$, our algorithm solves the problem in $\widetilde{O}(r\sqrt{mn}/\epsilon + \mathrm{poly}(n,1/\epsilon))$ quantum time, demonstrating both the applicability and the efficiency of quantum computing in accelerating regression tasks.

Regression is one of the most common ways to learn from data. It helps scientists, engineers, and economists understand how different quantities are related and make predictions from observations. However, when a dataset contains a very large number of data points, solving regression problems can become computationally expensive. Previous quantum algorithms mainly focused on the simplest forms of regression, leaving open whether quantum computers could also help with more flexible regression methods used in data analysis. This paper develops a quantum approach for speeding up a broad family of regression tasks. The main idea is to replace a large dataset with a much smaller weighted summary that still preserves the important information needed for the regression problem. Solving the problem on this summary gives nearly the same answer as solving it on the full dataset, but can be faster for large datasets. Our method applies to several common regression models, including methods designed to stabilize predictions, select important variables, or handle outliers. Under standard assumptions about how data is accessed by a quantum computer, the algorithm can significantly reduce the dependence on the number of data points, giving up to a quadratic speedup in some regimes. The result suggests that quantum algorithms may accelerate a broad range of basic data-analysis tasks, not just isolated special cases.