Authors’ summary: It is well known that the sparse grid algorithm has been widely accepted as an efficient tool to overcome the “curse of dimensionality” in some degree. In this note, we first give the error estimate of hyperbolic cross (HC) approximations with generalized Hermite functions. The exponential convergence in both regular and optimized HC approximations is shown. Moreover, the error estimate of Hermite spectral method to high-dimensional linear parabolic partial differential equations (PDEs) with HC approximations is investigated in the properly weighted Korobov spaces. The numerical result verifies the exponential convergence of this approach.