Generalized quantum singular value transformation with application in quantum conjugate gradient least squares algorithm
This work provides a more flexible quantum algorithm for matrix function implementation, potentially improving quantum linear algebra for researchers in quantum computing.
The authors extend generalized quantum signal processing (GQSP) to general matrices, creating generalized quantum singular value transformation (GQSVT), which relaxes parity constraints compared to QSVT. They then propose a hybrid quantum conjugate gradient least squares algorithm using GQSVT.
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.