Universal Matrix Multiplication on Quantum Computer

TL;DR

This work targets the matrix multiplication bottleneck in pattern recognition and ML by proposing a universal quantum matrix multiplication framework that leverages the Quantum Fourier Transform (QFT) for data encoding and optimized QFT-based adders and multipliers. It further extends the approach to a quantum Strassen algorithm, and provides resource estimates and IBM Q simulator experiments to demonstrate acceleration under favorable conditions, while clarifying trade-offs in qubit usage and circuit depth. Classical bounds on matrix multiply time, $O(n^{\omega+o(1)})$ with $2\le \omega \le 3$, Strassen's $O(n^{\log_2 7})$, and recent refinements (e.g., around $2.371866$) provide context for potential quantum gains. The work establishes a concrete pathway toward universal quantum matrix multiplication and highlights the practical prospects for speeding up ML model training on future fault-tolerant hardware or via quantum-classical hybrids on NISQ devices.

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.

Figures (8)

• Figure 1: The circuit for the Quantum Fourier Transform.
• Figure 2: Quantum circuits for adders; (a) The circuit for the original quantum adder; (b) The circuit for the optimized quantum adder.
• Figure 3: Quantum circuits for multipliers; (a) The circuit for the original quantum multiplier; (b) The circuit for the optimized quantum multiplier.
• Figure 4: Classical circuits for adder and multiplier; (a) Column multiplication; (d) Classical circuits of serial adder and array multiplier.
• Figure 5: Measurement results of the quantum simulation for the matrix product $C=AB$, where $A=\left[1234 \right]$ and $B=\left[2345 \right]$. (The results are in hexadecimal and represented in reverse order.)