  By using a different quantum-to-classical mapping from the Trotter-Suzuki
decomposition, we identify the entanglement structure of the maximal
eigenvectors for the associated quantum transfer matrix. This observation
provides a deeper insight into the problem of linear growth of the entanglement
entropy in time evolution using conventional methods. Based on this
observation, we propose a general method for arbitrary temperatures using the
biorthonormal transfer-matrix renormalization group. Our method exhibits a
competitive accuracy with a much cheaper computational cost in comparison with
two recent proposed methods for long-time dynamics based on a folding algorithm
[Phys. Rev. Lett. 102, 240603 (2009)] and a modified time-dependent
density-matrix renormalization group [Phys. Rev. Lett. 108, 227206 (2012)].
