# Generalized quantum singular value transformation with application in quantum conjugate gradient least squares algorithm

## TL;DR

This paper extends generalized quantum signal processing to general matrices, creating GQSVT, and applies it to develop a hybrid quantum-classical conjugate gradient least squares algorithm.

## Contribution

It introduces GQSVT, relaxing polynomial parity restrictions, and demonstrates its application in a hybrid quantum-classical algorithm for least squares problems.

## Key findings

- GQSVT relaxes parity restrictions compared to QSVT.
- The proposed hybrid algorithm applies GQSVT to solve least squares.
- Extension of GQSP to general matrices broadens quantum signal processing capabilities.

## Abstract

Quantum signal processing (QSP) and generalized quantum signal processing (GQSP) are essential tools for implementing the block encoding of matrix functions. The achievable polynomials of QSP have restrictions on parity, while GQSP eliminates these restrictions. But GQSP only constructs functions of unitary matrices. In this paper, we further investigate GQSP and extend it to general matrices. Compared with the quantum singular value transformation (QSVT), our proposed method relaxes the requirements on the parity of polynomials. We refer to this extension as generalized quantum singular value transformation (GQSVT). Subsequently, by utilizing the relationship between generalized matrix functions and standard matrix functions, we propose a classical-quantum hybrid quantum conjugate gradient least squares (CGLS) algorithm using GQSVT.