  In this paper we propose a quantum algorithm to determine the Tikhonov
regularization parameter and solve the ill-conditioned linear equations, for
example, arising from the finite element discretization of linear or nonlinear
inverse problems. For regularized least squares problem with a fixed
regularization parameter, we use the HHL algorithm and work on an extended
matrix with smaller condition number. For the determination of the
regularization parameter, we combine the classical L-curve and GCV function,
and design quantum algorithms to compute the norms of regularized solution and
the corresponding residual in parallel and locate the best regularization
parameter by Grover's search. The quantum algorithm can achieve a quadratic
speedup in the number of regularization parameters and an exponential speedup
in the dimension of problem size.
