Block-encoding is a standard framework for embedding matrices into unitary operators in quantum algorithms. Efficient implementation of products between block-encoded matrices is crucial for applications such as Hamiltonian simulation and quantum linear algebra. We present resource-efficient methods for matrix-matrix, Kronecker, and Hadamard products between block-encodings that apply to rectangular matrices of arbitrary dimensions. Our constructions significantly reduce the number of ancilla qubits, achieving exponential qubit savings for sequences of matrix-matrix multiplications, with a moderate increase in gate complexity. These product operations also enable more complex block-encodings, including a compression gadget for time-dependent Hamiltonian simulation and matrices represented as sums of Kronecker products, each with improved resource requirements.
@article{arxiv.2509.15779,
title = {Products between block-encodings},
author = {Dekuan Dong and Yingzhou Li and Jungong Xue},
journal= {arXiv preprint arXiv:2509.15779},
year = {2025}
}