  We investigate for which resource states an efficient classical simulation of
measurement based quantum computation is possible. We show that the
Schmidt--rank width, a measure recently introduced to assess universality of
resource states, plays a crucial role in also this context. We relate
Schmidt--rank width to the optimal description of states in terms of tree
tensor networks and show that an efficient classical simulation of measurement
based quantum computation is possible for all states with logarithmically
bounded Schmidt--rank width (with respect to the system size). For graph states
where the Schmidt--rank width scales in this way, we efficiently construct the
optimal tree tensor network descriptions, and provide several examples. We
highlight parallels in the efficient description of complex systems in quantum
information theory and graph theory.
