  The strong convergence rate of the Euler scheme for SDEs driven by additive
fractional Brownian motions is studied, where the fractional Brownian motion
has Hurst parameter $H\in(\frac13,\frac12)$ and the drift coefficient is not
required to be bounded. The Malliavin calculus, the rough path theory and the
$2$D Young integral are utilized to overcome the difficulties caused by the low
regularity of the fractional Brownian motion and the unboundedness of the drift
coefficient. The Euler scheme is proved to have strong order $2H$ for the case
that the drift coefficient has bounded derivatives up to order three and have
strong order $H+\frac12$ for linear cases. Numerical simulations are presented
to support the theoretical results.
