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 Gram-Schmidt process is a widely used method to solve these two problems. However, the existing methods, including Gram-Schmidt process have problems of high complexity, scaling $O(N^3)$ 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 Gram-Schmidt process and quantum phase estimation. The complexity of proposed quantum algorithms is also theoretically and numerically analyzed. We find that our algorithms provide polynomial acceleration over the best-known classical and quantum algorithms on these two problems, scaling $O(N^2\mathrm{poly}(\log N))$ in the dimension $N$ of the system.