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

TL;DR

The paper introduces a Quantum Kernel-based Matrix Multiplication (QKMM) algorithm that achieves an asymptotically optimal quantum time complexity of $O(N^2 \log_2 N)$, surpassing the best classical bound of $O(N^{2.371552})$. It builds a hierarchical framework of quantum circuits—V 2V, V 2M, M 2M, and M-MM—based on amplitude-encoded data and quantum kernels to perform inner products and matrix products within a single coherent circuit. Complexity is analyzed using a quantum-gate-count metric, and the authors provide an explicit gate-count formula, along with simulation results in noiseless and noisy settings to validate scalability and robustness on near-term hardware. The work also discusses the need for improved data-encoding strategies to address noise and depth challenges, positioning quantum-intensive computing as a pathway to accelerate compute-heavy tasks in AI and scientific computing.

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.

Figures (6)

• Figure 1: Comparison of time complexity between quantum matrix multiplication algorithm and classical algorithms.
• Figure 2: Time consumption analysis of matrix multiplication based on different quantum circuits; (a) Comparison of V 2V computational efficiency; (b) Comparison of V 2M computational efficiency; (c) Comparison of M 2M computational efficiency; (d) Comparison of M-MM computational efficiency.
• Figure 3: Performance comparison of quantum kernel-based inner product algorithms and classical approaches (Hadamard Test and Swap Test) under diverse noise models; (a) Scaling trends of fidelity against dimensionality under noiseless, single-source ($T_1$, $T_2$, gate noise), and composite noise environments; (b) Cumulative analysis of mean error under corresponding noise conditions.
• Figure 4: Robustness analysis of the QKMM algorithm in multi-noise environments; (a) Fidelity performance as a function of dimensional scaling under single-source ($T_1$, $T_2$, gate noise) and composite noise; (b) Evaluation of mean error under corresponding noise conditions.
• Figure 5: Quantum circuits design for the QKMM algorithm suite; (a) The quantum circuit for V 2V; (b) The quantum circuit for V 2M; (c) The quantum circuit for M 2M; (d) The quantum circuit for M-MM.