  We prove that conformal curved spacetime can be encoded into the initial wave
function and that curved propagation can be simulated on a two-dimensional
regular lattice with a finite set of homogeneous unitary operators. We
generalize recent results shown in [1], where the author transforms
flat-spacetime in curved static spacetime via non-unitary rotations. In
particular, encoding the metric in the initial quantum state via unitary local
rotations, can naturally be useful for quantum simulators due to the efficiency
and simplicity of the implementation scheme. We validate our model proving that
the continuous limit converges to the Dirac Eq. in curved spacetime.
