  Non-Markovian open quantum systems represent the most general dynamics when
the quantum system is coupled with a bath environment. The quantum dynamics
arising from many important applications are non-Markovian. Although for
special cases, such as Hamiltonian evolution and Lindblad evolution, quantum
simulation algorithms have been extensively studied, efficient quantum
simulations for the dynamics of non-Markovian open quantum systems remain
underexplored. The most immediate obstacle for studying such systems is the
lack of a universal succinct description of their dynamics. In this work, we
fulfill the gap of studying such dynamics by 1) providing a succinct
representation of the dynamics of non-Markovian open quantum systems with
quantifiable error, and 2) developing an efficient quantum algorithm for
simulating such dynamics with cost $O(t\, \mathrm{polylog}(t/\epsilon))$ for
evolution time $t$ and precision $\epsilon$. Our derivation of the succinct
representation is based on stochastic Schr\"odinger equations, which could lead
to new alternatives to deal with open quantum systems as well.
