  The high-index saddle dynamics (HiSD) method [J. Yin, L. Zhang, and P. Zhang,
{\it SIAM J. Sci. Comput., }41 (2019), pp.A3576-A3595] serves as an efficient
tool for computing index-$k$ saddle points and constructing solution
landscapes. Nevertheless, the conventional HiSD method often encounters slow
convergence rates on ill-conditioned problems. To address this challenge, we
propose an accelerated high-index saddle dynamics (A-HiSD) by incorporating the
heavy ball method. We prove the linear stability theory of the continuous
A-HiSD, and subsequently estimate the local convergence rate for the discrete
A-HiSD. Our analysis demonstrates that the A-HiSD method exhibits a faster
convergence rate compared to the conventional HiSD method, especially when
dealing with ill-conditioned problems. We also perform various numerical
experiments including the loss function of neural network to substantiate the
effectiveness and acceleration of the A-HiSD method.
