Is the efficiency of classical simulations of quantum dynamics related to integrability?

Efficiency of time evolution of quantum observables, and thermal states of quenched Hamiltonians, is studied using time-dependent density-matrix renormalization group method in a family of generic quantum spin chains which undergo a transition from integrable to nonintegrable-quantum chaotic case as control parameters are varied. Quantum states (observables) are represented in terms of matrix-product operators with rank D-epsilon(t), such that evolution of a long chain is accurate within fidelity error epsilon up to time t. We found that the rank generally increases exponentially D-epsilon(t)proportional to exp(const t), unless the model system was integrable in which case we found polynomial increase.