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.

Quantum Physics (quant-ph); Computational Complexity (cs.CC); Machine Learning (cs.LG)

arXiv:2602.05541 [quant-ph]

(or arXiv:2602.05541v3 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2602.05541 Focus to learn more arXiv-issued DOI via DataCite

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.