  An infinite projected entangled pair state (iPEPS) is a tensor network ansatz
to represent a quantum state on an infinite 2D lattice whose accuracy is
controlled by the bond dimension $D$. Its real, Lindbladian or imaginary time
evolution can be split into small time steps. Every time step generates a new
iPEPS with an enlarged bond dimension $D' > D$, which is approximated by an
iPEPS with the original $D$. In Phys. Rev. B 98, 045110 (2018) an algorithm was
introduced to optimize the approximate iPEPS by maximizing directly its
fidelity to the one with the enlarged bond dimension $D'$. In this work we
implement a more efficient optimization employing a local estimator of the
fidelity. For imaginary time evolution of a thermal state's purification, we
also consider using unitary disentangling gates acting on ancillas to reduce
the required $D$. We test the algorithm simulating Lindbladian evolution and
unitary evolution after a sudden quench of transverse field $h_x$ in the 2D
quantum Ising model. Furthermore, we simulate thermal states of this model and
estimate the critical temperature with good accuracy: $0.1\%$ for $h_x=2.5$ and
$0.5\%$ for the more challenging case of $h_x=2.9$ close to the quantum
critical point at $h_x=3.04438(2)$.
