A generic of many systems require complex description, however the states appearing in physical systems not necessary so. These often satisfy so called area law for entropy, or equivalently are approximated by low complexity Matrix Product States aka Finitely Correlated States.
In KCIK we have proved that exponential decay of correlations of a state implies area law ergo simple description of the state (see ).Simulating random evolution
One can ask how complex is to simulate random unitary evolution on many body system by random quantum gates (two-body unitaries). In  we have shown that such random unitary can be efficiently simulated, i.e. the length of a circuit needed to approximate a given moment of Haar measure is polynomial in number of systems and in the moment. F.G.S.L. Brandão, M. Horodecki, An area law for entanglement from exponential decay of correlations, Nat. Phys. 9, 721, (2013).
 F.G.S.L. Brandao, A.W. Harrow, M. Horodecki, Local random quantum circuits are approximate polynomial-designs, Preprint at https://arxiv.org/abs/1208.0692 (2015).