  We present a new method to accelerate real time-time dependent density
functional theory (rt-TDDFT) calculations with hybrid exchange-correlation
functionals. For large basis set, the computational bottleneck for large scale
calculations is the application of the Fock exchange operator to the
time-dependent orbitals. Our main goal is to reduce the frequency of applying
the Fock exchange operator, without loss of accuracy. We achieve this by
combining the recently developed parallel transport (PT) gauge formalism and
the adaptively compressed exchange operator (ACE) formalism. The PT gauge
yields the slowest possible dynamics among all choices of gauge. When coupled
with implicit time integrators such as the Crank-Nicolson (CN) scheme, the
resulting PT-CN scheme can significantly increase the time step from
sub-attoseconds to 10-100 attoseconds. At each time step $t_{n}$, PT-CN
requires the self-consistent solution of the orbitals at time $t_{n+1}$. We use
ACE to delay the update of the Fock exchange operator in this nonlinear system,
while maintaining the same self-consistent solution. We verify the performance
of the resulting PT-CN-ACE method by computing the absorption spectrum of a
benzene molecule and the response of bulk silicon systems to an ultrafast laser
pulse, using the planewave basis set and the HSE functional. We report the
strong and weak scaling of the PT-CN-ACE method for silicon systems ranging
from 32 to 1024 atoms, with up to 2048 computational cores. Compared to
standard explicit time integrators such as the 4th order Runge-Kutta method
(RK4), the PT-CN-ACE can reduce the Fock exchange operator application by
nearly 70 times, thus reduce the overall wall clock time time by 46 times for
the system with 1024 atoms. Hence our work enables hybrid functional rt-TDDFT
calculations to be routinely performed with a large basis set for the first
time.
