  Two models are first presented, of one-dimensional discrete-time quantum walk
(DTQW) with temporal noise on the internal degree of freedom (i.e., the coin):
(i) a model with both a coin-flip and a phase-flip channel, and (ii) a model
with random coin unitaries. It is then shown that both these models admit a
common limit in the spacetime continuum, namely, a Lindblad equation with
Dirac-fermion Hamiltonian part and, as Lindblad jumps, a chirality flip and a
chirality-dependent phase flip, which are two of the three standard error
channels for a two-level quantum system. This, as one may call it, Dirac
Lindblad equation, provides a model of quantum relativistic spatial diffusion,
which is evidenced both analytically and numerically. This model of spatial
diffusion has the intriguing specificity of making sense only with original
unitary models which are relativistic in the sense that they have chirality, on
which the noise is introduced: The diffusion arises via the by-construction
(quantum) coupling of chirality to the position. For a particle with vanishing
mass, the model of quantum relativistic diffusion introduced in the present
work, reduces to the well-known telegraph equation, which yields propagation at
short times, diffusion at long times, and exhibits no quantumness. Finally, the
results are extended to temporal noises which depend smoothly on position.
