  A discrete-time Quantum Walk (QW) is essentially an operator driving the
evolution of a single particle on the lattice, through local unitaries. Some
QWs admit a continuum limit, leading to familiar PDEs (e.g. the Dirac
equation). Recently it was discovered that prior grouping and encoding allows
for more general continuum limit equations (e.g. the Dirac equation in $(1+1)$
curved spacetime). In this paper, we extend these results to arbitrary space
dimension and internal degree of freedom. We recover an entire class of PDEs
encompassing the massive Dirac equation in $(3+1)$ curved spacetime. This means
that the metric field can be represented by a field of local unitaries over a
lattice.
