  Tensor network methods as presented in our open source Matrix Product States
software have opened up the possibility to study many-body quantum physics in
one and quasi-one-dimensional systems in an easily accessible package similar
to density functional theory codes but for strongly correlated dynamics. Here,
we address methods which allow one to capture the full entanglement without
truncation of the Hilbert space. Such methods are suitable for validation of
and comparisons to tensor network algorithms, but especially useful in the case
of new kinds of quantum states with high entanglement violating the truncation
in tensor networks. Quantum cellular automata are one example for such a
system, characterized by tunable complexity, entanglement, and a large spread
over the Hilbert space. Beyond the evolution of pure states as a closed system,
we adapt the techniques for open quantum systems simulated via the Lindblad
master equation. We present three algorithms for solving closed-system
many-body time evolution without truncation of the Hilbert space. Exact
diagonalization methods have the advantage that they not only keep the full
entanglement but also require no approximations to the propagator. Seeking the
limits of a maximal number of qubits on a single core, we use Trotter
decompositions or Krylov approximation to the exponential of the Hamiltonian.
All three methods are also implemented for open systems represented via the
Lindblad master equation built from local channels. We show their convergence
parameters and focus on efficient schemes for their implementations including
Abelian symmetries, e.g., U(1) symmetry used for number conservation in the
Bose-Hubbard model or discrete Z2 symmetries in the quantum Ising model. We
present the thermalization timescale in the long-range quantum Ising model as a
key example of how exact diagonalization contributes to novel physics.
