报告题目：Quantum Numerical Linear Algebra
报告摘要：We will review the development of quantum algorithms, especially the HHL for solving linear algebraic systems, then switch to our quantum algorithms on the numerical linear algebra problems, including the circulant preconditioner, regularized least squares (LS), total least squares (TLS), and tridiagonal eigensolvers. (1) We consider the quantum linear solver for Ax = b with the circulant preconditioner. The main technique is the modified singular value estimation (SVE). (2) The regularized LS can be used to solve an ill-conditioned problem. The determination of the proper regularization parameter is the key step. Combining the L-curve or the Hanke-Raus rule with the HHL and quantum amplitude estimation, we propose quantum algorithms to compute the norms of regularized solution and the corresponding residual, then locate the best regularization parameter by Grover's algorithm. This yields a quadratic speedup in the number of regularization parameters. (3) For the TLS problem, it is transformed to finding the ground state of a Hamiltonian matrix. We propose quantum algorithms for solving this problem based on quantum simulation of resonant transitions. Our algorithms can achieve at least polynomial speedup over the known classical algorithms.