  Recently in [Phys. Rev. B 99, 241301(R) (2019)] tensor networks build upon
logical circuits were briefly introduced to retrieve exciton and biexciton
states. Compared to a conventional approach the tensor network methods scales
logarithmic instead of linear in the grid points of the Brioullin zone and
linear instead of exponential in the number of electrons and holes. This
enables calculations with higher precision on the full Brioullin zone than
previously possible. In this paper extensive details for an efficient
implementation and the corresponding mathematical background are presented. In
particular this includes applications and results for excitons, trions and
biexcitons (for monolayer MoS$_2$ as example), going beyond the initial brief
introduction. Furthermore strategies for calculating selective excited bound
states and tests of common approximations are discussed making use of the high
accuracy full Brioullin zone treatment of the tensor network method.
