The Trotter-Suzuki approximation leads to an efficient algorithm for solving the timedependent Schrödinger equation. Using existing highly optimized CPU and GPU kernels, we developed a distributed version of the algorithm that runs efficiently on a cluster. Our implementation also improves single node performance, and is able to use multiple GPUs within a node. The scaling is close to linear using the CPU kernels, whereas the efficiency of GPU kernels improve with larger matrices. We also introduce a hybrid kernel that simultaneously uses multicore CPUs and GPUs in a distributed system. This kernel is shown to be efficient when the matrix size would not fit in the GPU memory. Larger quantum systems scale especially well with a high number nodes. The code is available under an open source license. This work was carried out while P. W. was visiting the Department of Computer Applications in Science & Engineering at the Barcelona Supercomputing Center, funded by the \Access to BSC Facilities" project of the HPC-Europe2 programme (contract no. 228398). Peer Reviewed