  We extend the concept of dual unitary quantum gates to quantum lattice models
in $2 + 1$ dimensions, by introducing and studying ternary unitary
four-particle gates, which are unitary in time and both spatial dimensions.
When used as building blocks of lattice models with periodic boundary
conditions in time and space (corresponding to infinite temperature states),
dynamical correlation functions exhibit a light-ray structure. We also
generalize solvable MPS to two spatial dimensions with cylindrical boundary
conditions, by showing that the analogous solvable PEPS can be identified with
matrix product unitaries. In the resulting tensor network for evaluating
equal-time correlation functions, the bulk ternary unitary gates cancel out. We
delineate and implement a numerical algorithm for computing such correlations
by contracting the remaining tensors.
