  The simple update (SU) and full update (FU) are the two paradigmatic time
evolution algorithms for a tensor network known as the infinite projected
entangled pair state (iPEPS). They differ by an error measure that is either,
respectively, local or takes into account full infinite tensor environment. In
this paper we test an intermediate neighborhood tensor update (NTU) accounting
for the nearest neighbor environment. This small environment can be contracted
exactly in a parallelizable way. It provides an error measure that is Hermitian
and non-negative down to machine precision. In the 2D quantum Ising model NTU
is shown to yield stable unitary time evolution following a sudden quench. It
also yields accurate thermal states despite correlation lengths that reach up
to 20 lattice sites. The latter simulations were performed with a manifestly
Hermitian purification of a thermal state. Both were performed with reduced
tensors that do not include physical (and ancilla) indices. This modification
naturally leads to two other schemes: a local SVD update (SVDU) and a full
tensor update (FTU) being a variant of FU.
