Reducing the Complexity of Matrix Multiplication by Quantum Computing

Abstract

Matrix multiplication is a fundamental operation in compute-intensive tasks and a key component of modern quantum acceleration frameworks. Here we present a quantum matrix multiplication algorithm based on quantum kernels (QKMM), achieving an elementary gate complexity of \(O(N^2\log_2N)\), with amplitude encoding overhead explicitly included and without assuming a QRAM oracle. This scaling is asymptotically lower than that of the best-known classical matrix multiplication algorithm \(O(N^{2.371339})\). Building upon QKMM, we establish a family of quantum linear algebra operators, including Quantum Vector Inner Product (V${\scriptstyle 2}$V), Quantum Vector-Matrix Multiplication (V${\scriptstyle 2}$M), QKMM (M${\scriptstyle 2}$M), Quantum One-to-Many Matrix Multiplication(O${\scriptstyle 2}$M) and Quantum Sequential Matrix Multiplication (SMM), providing a unified framework from vector operations to parallel and sequential matrix transformations. Through noiseless simulations, realistic noise modelling and experiments on a superconducting quantum processor, we systematically characterize the numerical accuracy, resource requirements and hardware execution limits of this operator framework. Furthermore, we integrate SMM into deep neural-network inference, enabling intermediate features to propagate coherently across layers without repeated measurement and re-encoding. These results establish a pathway from quantum circuit-level algorithm design to end-to-end coherent computation, providing a quantum computing framework for matrix-centric compute-intensive applications.