  We introduce a numerical algorithm to simulate the time evolution of a matrix
product state under a long-ranged Hamiltonian. In the effectively
one-dimensional representation of a system by matrix product states,
long-ranged interactions are necessary to simulate not just many physical
interactions but also higher-dimensional problems with short-ranged
interactions. Since our method overcomes the restriction to short-ranged
Hamiltonians of most existing methods, it proves particularly useful for
studying the dynamics of both power-law interacting one-dimensional systems,
such as Coulombic and dipolar systems, and quasi two-dimensional systems, such
as strips or cylinders. First, we benchmark the method by verifying a
long-standing theoretical prediction for the dynamical correlation functions of
the Haldane-Shastry model. Second, we simulate the time evolution of an
expanding cloud of particles in the two-dimensional Bose-Hubbard model, a
subject of several recent experiments.
