<p>We consider the problem of the optimal choice of regularization parameter in the stabilization scheme of the finite-element method for a singularly perturbed diffusion–advection–reaction problem. Stabilization is based on the combination of a Tikhonov-type regularization with an auxiliary Cauchy problem. We study the behavior of perturbations in the approximate solution depending on changes in the regularization parameter. On the basis of the performed analysis, we construct an heuristic criterion for the optimal choice of the regularization parameter. The criterion is formulated as a local problem of minimization of the corresponding function constructed as a composition of a linear functional and the obtained finite-element approximation. The proposed approach was originally developed for one-dimensional problems and then generalized to the case of 2<i>D</i> problems. We also discuss the possibility of application of the Harrow–Hassidim–Lloyd quantum algorithm in combination with the swap test in order to implement the evaluation of the obtained loss function on quantum computers.</p>