Quantum Communication Complexity of L2-Regularized Linear Regression Protocols

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.