Quantum block encoding for semiseparable matrices

TL;DR

This paper introduces a novel quantum block encoding method specifically for one-pair semiseparable matrices, utilizing matrix factorization to efficiently embed these matrices into unitary matrices with minimal qubits.

Contribution

It presents a new quantum block encoding technique for rank-structured matrices, focusing on one-pair semiseparable matrices, with efficient resource usage and error bounds.

Findings

Requires $2\log(N)+7$ ancillary qubits

Polylogarithmic encoding time

Error bound of $\\mathcal{O}(N^2)$

Abstract

Quantum block encoding (QBE) is a crucial step in the development of most quantum algorithms, as it provides an embedding of a given matrix into a suitable larger unitary matrix. Historically, the development of efficient techniques for QBE has mostly focused on sparse matrices; less effort has been devoted to data-sparse (e.g., rank-structured) matrices. In this work we examine a particular case of rank structure, namely, one-pair semiseparable matrices. We present a new block encoding approach that relies on a suitable factorization of the given matrix as the product of triangular and diagonal factors. To encode the matrix, the algorithm needs $2\log(N)+7$ ancillary qubits. This process takes polylogarithmic time and has an error of $\mathcal{O}(N^2)$, where $N$ is the matrix size.