Is 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 non-integrable - 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 find that rank generally increases exponentially, D_\epsilon(t) \propto \exp(const t), unless the system is integrable in which case we find polynomial increase.