Reducing the Complexity of Matrix Multiplication to $O(N^2log_2N)$ by an Asymptotically Optimal Quantum Algorithm

TL;DR

This paper introduces a quantum algorithm for matrix multiplication that achieves near-optimal asymptotic complexity, outperforming classical methods and demonstrating practical advantages through simulations.

Contribution

It presents a novel quantum kernel-based matrix multiplication algorithm with asymptotically optimal complexity of O(N^2 log N), improving over classical algorithms.

Findings

The quantum algorithm achieves asymptotic complexity of O(N^2 log N).

Simulation results show practical runtime and stability benefits.

The approach outperforms classical matrix multiplication in theory and practice.

Abstract

Matrix multiplication is a fundamental classical computing operation whose efficiency becomes a major challenge at scale, especially for machine learning applications. Quantum computing, with its inherent parallelism and exponential storage capacity, offers a potential solution to these limitations. This work presents a quantum kernel-based matrix multiplication algorithm (QKMM) that achieves an asymptotically optimal computational complexity of $ O(N^2 \log_2 N) $, outperforming the classical optimal complexity of $ O(N^{2.371552}) $, where $N$ denotes the matrix dimension. Through noiseless and noisy quantum simulation experiments, we demonstrate that the proposed algorithm not only exhibits superior theoretical efficiency but also shows practical advantages in runtime performance and stability.