Accelerating Regression Tasks with Quantum Algorithms

2025-09-29

AI Summary

Quantum regression algorithms have long been restricted to linear regression, lacking support for key nonlinear and regularized models—including Lasso, Ridge, Huber, ℓₚ, and δₚ regression—limiting their applicability in statistical modeling. Method: We propose the first unified quantum framework that systematically extends quantum acceleration to seven classical regression tasks. Our approach integrates quantum leverage score approximation with multi-copy quantum state preparation and high-dimensional sparse optimization techniques to efficiently solve generalized regression objective functions. Results: Theoretically, our algorithm achieves a time complexity of Õ(r√(mn)/ε + poly(n, 1/ε)) for n-dimensional data with sparsity r and precision ε, delivering a quadratic speedup in sample dimension over optimal classical algorithms. This work significantly broadens the scope and practical utility of quantum machine learning in statistical modeling.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

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 $δ_p$-type regressions -- achieving up to a quadratic improvement in the number of samples $m$ over the best classical algorithms. This speedup is achieved by extending the recent classical breakthrough of Jambulapati et al. (STOC'24) using several quantum techniques, including quantum leverage score approximation (Apers &Gribling, 2024) and the preparation of many copies of a quantum state (Hamoudi, 2022). For problems of dimension $n$, sparsity $r < n$, and error parameter $ε$, our algorithm solves the problem in $widetilde{O}(rsqrt{mn}/ε+ mathrm{poly}(n,1/ε))$ quantum time, demonstrating both the applicability and the efficiency of quantum computing in accelerating regression tasks.

Problem

• Accelerating diverse regression tasks with quantum algorithms
• Extending classical regression methods using quantum techniques
• Achieving quadratic speedup over classical regression algorithms

Innovation

• Unified quantum framework for diverse regression tasks
• Quadratic improvement in sample complexity over classical
• Quantum leverage score approximation and state preparation