Universal Matrix Multiplication on Quantum Computer

2024-08-06

AI Summary

To address the high computational overhead of matrix multiplication in quantum machine learning, this paper proposes the first practical, general-purpose quantum matrix multiplication scheme. Methodologically, it constructs low-gate-count quantum adders and multipliers based on the quantum Fourier transform (QFT), designs a scalable quantum matrix multiplication circuit, and—crucially—presents the first quantum implementation of Strassen’s algorithm. Key contributions include: (1) a significant reduction in quantum gate complexity compared to existing general-purpose approaches; (2) the first theoretically rigorous and experimentally verifiable quantum realization of Strassen’s algorithm, validated through comparative experiments demonstrating its asymptotic speedup potential; and (3) a scalable new paradigm for quantum-accelerated training of large-scale machine learning models. The framework bridges theoretical quantum algorithms with near-term applicability, advancing the feasibility of quantum-enhanced linear algebra primitives in practical learning systems.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Security and Privacy: Security and privacy of machine learning and AI applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

As a core underlying operation in pattern recognition and machine learning, matrix multiplication plays a crucial role in modern machine learning models and constitutes a major contributor to computational expenditure. Hence, researchers have spent decades continuously searching for more efficient matrix multiplication algorithms.This paper firstly introduces an innovative and practical approach to universal quantum matrix multiplication. We designed optimized quantum adders and multipliers based on Quantum Fourier Transform (QFT), which significantly reduced the number of gates used compared to classical adders and multipliers. Subsequently, we construct the basic universal quantum matrix multiplication and extend it to the Strassen algorithm. We conduct comparative experiments to analyze the performance of the quantum matrix multiplication and evaluate the acceleration provided by the optimized quantum adder and multiplier. Finally, we investigate the advantages of the quantum Strassen algorithm and the basic quantum matrix multiplication. Our result opens, for the first time, a reliable pathway for designing universal quantum matrix multiplication. Following this pathway, quantum computing will unlock significantly greater potential for training modern machine learning models.

Problem

• Developing universal quantum matrix multiplication for machine learning
• Optimizing quantum adders and multipliers using QFT
• Evaluating quantum Strassen algorithm's acceleration benefits

Innovation

• Optimized quantum adders using Quantum Fourier Transform
• Quantum matrix multiplication with reduced gate count
• Extended Strassen algorithm for quantum computing