# Reducing the Complexity of Matrix Multiplication by Quantum Computing

arXiv: https://arxiv.org/abs/2602.05541 · HTML: https://mlsanity.com/paper/2602.05541/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 21 references that are in the map.
- First posted on arXiv 5 Feb 2026.

## Builds on (most cited first, 6 of 6)

- [Supervised learning with quantum enhanced feature spaces](https://mlsanity.com/paper/1804.11326/) (2018)
- [A Refined Laser Method and Faster Matrix Multiplication](https://mlsanity.com/paper/2010.05846/) (2020)
- [New Bounds for Matrix Multiplication: from Alpha to Omega](https://mlsanity.com/paper/2307.07970/) (2023)
- [Parallel Multi Channel Convolution using General Matrix Multiplication](https://mlsanity.com/paper/1704.04428/) (2017)
- [More Asymmetry Yields Faster Matrix Multiplication](https://mlsanity.com/paper/2404.16349/) (2024)
- [Faster Matrix Multiplication via Asymmetric Hashing](https://mlsanity.com/paper/2210.10173/) (2022)

## Abstract (arXiv)

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(N2log2N)), 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(N2.371339)). Building upon QKMM, we establish a family of quantum linear algebra operators, including Quantum Vector Inner Product (Vscriptstyle 2V), Quantum Vector-Matrix Multiplication (Vscriptstyle 2M), QKMM (Mscriptstyle 2M), Quantum One-to-Many Matrix Multiplication(Oscriptstyle 2M) 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.