Quantum Singular Value Decomposition Based on QR Iteration

Aiming at the high computational complexity of classical algorithms for the singular value decomposition (SVD) of large matrices, as well as the limitations of existing quantum SVD algorithms that require the matrix to be decomposed to possess non-sparse, low-rank characteristics, and the difficulty of constructing unitary matrices of arbitrary size for current quantum computers, a quantum SVD algorithm based on QR iteration was introduced. QR iteration is a numerical algorithm for calculating the eigenvalues and eigenvectors of matrices. The QR iteration leverages Householder transformations to execute the classical matrix multiplication operations via quantum matrix multiplication computations. Results show that the proposed approach can obtain the singular values and singular matrices of the target matrix, and it is feasible to perform SVD on large-scale matrices.