  Matrix Product Operators (MPOs) are at the heart of the second-generation
Density Matrix Renormalisation Group (DMRG) algorithm formulated in Matrix
Product State language. We first summarise the widely known facts on MPO
arithmetic and representations of single-site operators. Second, we introduce
three compression methods (Rescaled SVD, Deparallelisation and Delinearisation)
for MPOs and show that it is possible to construct efficient representations of
arbitrary operators using MPO arithmetic and compression. As examples, we
construct powers of a short-ranged spin-chain Hamiltonian, a complicated
Hamiltonian of a two-dimensional system and, as proof of principle, the
long-range four-body Hamiltonian from quantum chemistry.
