Reducing the Complexity of Matrix Multiplication by Quantum Computing

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.