Quantum Communication Complexity of L2-Regularized Linear Regression Protocols

TL;DR

This paper advances quantum distributed protocols for linear regression, significantly reducing communication complexity and precision requirements, especially for L2-regularized problems, through novel quantum techniques.

Contribution

It introduces improved quantum protocols for both ordinary and L2-regularized least squares, enhancing efficiency and extending applicability in distributed quantum computing.

Findings

Reduced quantum communication complexity for ordinary least squares

Quadratic improvement in precision digit requirements

First quantum protocol for L2-regularized least squares

Abstract

Linear regression is fundamental to statistical analysis and machine learning, but its application to large-scale datasets necessitates distributed computing. The problem also arises in quantum computing, where handling extensive data requires distributed approaches. This paper investigates distributed linear regression in the quantum coordinator model. Building upon the distributed quantum least squares protocol developed by Montanaro and Shao, we propose improved and extended quantum protocols for solving both ordinary (unregularized) and L2-regularized (Tikhonov) least squares problems. For ordinary least squares methods, our protocol reduces the quantum communication complexity compared to the previous protocol. In particular, this yields a quadratic improvement in the number of digits of precision required for the generated quantum states. This improvement is achieved by incorporating advanced techniques such as branch marking and branch-marked gapped phase estimation developed by Low and Su. Additionally, we introduce a quantum protocol for the L2-regularized least squares problem and derive its quantum communication complexity.