Lower bounds for quantum-inspired classical algorithms via communication complexity
Nikhil S. Mande, Changpeng Shao
TL;DR
This work develops a principled bridge between quantum-inspired classical algorithms and communication complexity to study lower bounds for matrix-related tasks. The authors introduce SQ access and a coordinator communication model, proving that efficient quantum-inspired algorithms imply low-communication protocols, which enables deriving lower bounds from hard communication problems like Set-Disjointness and Gap-Hamming. They establish quadratic lower bounds in the Frobenius norm for several tasks (linear regression, PCA, recommender systems, Hamiltonian simulation) and near-tight results for supervised clustering, revealing at least a quadratic quantum-classical separation for these problems. The framework extends to quantum query complexity for matrix problems via block-encoding and eigenvalue transformations, highlighting a broader connection between quantum algorithms and communication complexity with potential practical implications for distributed computation and ML-inspired tasks. Overall, the paper provides a versatile method to certify limits of quantum-inspired classical approaches and to guide the search for tighter lower bounds and more efficient quantum algorithms.
Abstract
Quantum-inspired classical algorithms provide us with a new way to understand the computational power of quantum computers for practically-relevant problems, especially in machine learning. In the past several years, numerous efficient algorithms for various tasks have been found, while an analysis of lower bounds is still missing. Using communication complexity, in this work we propose the first method to study lower bounds for these tasks. We mainly focus on lower bounds for solving linear regressions, supervised clustering, principal component analysis, recommendation systems, and Hamiltonian simulations. For those problems, we prove a quadratic lower bound in terms of the Frobenius norm of the underlying matrix. As quantum algorithms are linear in the Frobenius norm for those problems, our results mean that the quantum-classical separation is at least quadratic. As a generalisation, we extend our method to study lower bounds analysis of quantum query algorithms for matrix-related problems using quantum communication complexity. Some applications are given.