  In this paper, we consider the strong convergence order of the exponential
integrator for the stochastic heat equation driven by an additive fractional
Brownian motion with Hurst parameter $H\in(\frac12,1)$. By showing the strong
order one of accuracy of the exponential integrator under appropriote
assumptions, we present the first super-convergence result in temporal
direction on full discretizations for stochastic partial differential equations
driven by infinite dimensional fractional Brownian motions with Hurst parameter
$H\in(\frac12,1)$. The proof is a combination of Malliavin calculus, the
$L^p(\Omega)$-estimate of the Skorohod integral and the smoothing effect of the
Laplacian operator.
