Lower bounds for quantum-inspired classical algorithms via communication complexity

2024-02-24

arXiv.org

AI Summary

This work aims to rigorously characterize the computational limits of quantum-inspired classical algorithms—particularly those applied to matrix problems such as linear regression, principal component analysis (PCA), and recommendation systems—and to quantify the asymptotic speedup gap between them and genuine quantum algorithms. Method: We introduce the first unified analytical framework grounded in communication complexity, deriving tight lower bounds for both quantum-inspired classical algorithms (via classical communication complexity) and quantum query algorithms (via quantum communication complexity). Contribution/Results: We prove that, for several fundamental matrix tasks, any quantum-inspired classical algorithm must incur Ω(1/ε²) dependence on the input’s Frobenius norm to achieve ε-accuracy, whereas the corresponding quantum algorithms achieve O(1/ε) dependence. This establishes an unconditional quadratic separation, demonstrating that the quadratic speedup is inherent and unattainable by classical means—even those leveraging “quantum-inspired” sampling and sketching techniques. Our results provide the first information-theoretic foundation revealing the intrinsic limitations of the quantum-inspired paradigm.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Non-convex Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Quantum-inspired Algorithms
• Quantum Advantage
• Classical Computing Limits

Innovation

• Quantum-inspired Algorithms
• Communication Complexity
• Quantum Advantage