Quantum Algorithm for Vector Set Orthogonal Normalization and Matrix QR Decomposition with Polynomial Speedup

Vector set orthogonal normalization and matrix QR decomposition are fundamental problems in matrix analysis with important applications in many fields. We know that the Gram-Schmidt process is a widely used method to solve these two problems. However, existing methods, including the Gram-Schmidt process, have problems of high complexity, scaling O(N3) in the system dimension N, which leads to difficulties when calculating large-scale or ill-conditioned problems. With the development of quantum information processing, a series of quantum algorithms have been proposed, providing advantages and speedups over classical algorithms in many fields. In this paper we propose quantum algorithms to solve these two problems based on the idea of the Gram-Schmidt process and quantum phase estimation. The complexity of proposed quantum algorithms is theoretically and numerically analyzed. We find that our algorithms provide polynomial acceleration over the best-known quantum algorithms and a potential polynomial acceleration over the best-known classical algorithms on these two problems, scaling O(N2poly(logN)) in the dimension N of the system when ignoring the complexity of state readout or O(N2poly(logN)) when considering the complexity of state readout.