  We present an efficient quantum algorithm for simulating the dynamics of
Markovian open quantum systems. The performance of our algorithm is similar to
the previous state-of-the-art quantum algorithm, i.e., it scales linearly in
evolution time and poly-logarithmically in inverse precision. However, our
algorithm is conceptually cleaner, and it only uses simple quantum primitives
without compressed encoding. Our approach is based on a novel mathematical
treatment of the evolution map, which involves a higher-order series expansion
based on Duhamel's principle and approximating multiple integrals using scaled
Gaussian quadrature. Our method easily generalizes to simulating quantum
dynamics with time-dependent Lindbladians. Furthermore, our method of
approximating multiple integrals using scaled Gaussian quadrature could
potentially be used to produce a more efficient approximation of time-ordered
integrals, and therefore can simplify existing quantum algorithms for
simulating time-dependent Hamiltonians based on a truncated Dyson series.
