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Efficiency of time-evolution of quantum observables, and thermal states ofquenched hamiltonians, is studied using time-dependent density matrixrenormalization group method in a family of generic quantum spin chains whichundergo a transition from integrable to non-integrable - quantum chaotic caseas control parameters are varied. Quantum states (observables) are representedin terms of matrix-product-operators with rank D_\epsilon(t), such thatevolution of a long chain is accurate within fidelity error \epsilon up to timet. We find that rank generally increases exponentially, D_\epsilon(t) \propto\exp(const t), unless the system is integrable in which case we find polynomialincrease.Comment: 4 pages; v2. added paragraph discussing pure state

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