  A discrete-time Quantum Walk (QW) is essentially a unitary operator driving
the evolution of a single particle on the lattice. Some QWs admit a continuum
limit, leading to familiar PDEs (e.g. the Dirac equation). In this paper, we
study the continuum limit of a wide class of QWs, and show that it leads to an
entire class of PDEs, encompassing the Hamiltonian form of the massive Dirac
equation in $(1+1)$ curved spacetime. Therefore a certain QW, which we make
explicit, provides us with a unitary discrete toy model of a test particle in
curved spacetime, in spite of the fixed background lattice. Mathematically we
have introduced two novel ingredients for taking the continuum limit of a QW,
but which apply to any quantum cellular automata: encoding and grouping.
