A numerical scheme that solves the time-dependent Dirac equation is presented in which the time evolution is performed by an operator-splitting decomposition technique combined with the method ofcharacteristics. On a classical computer, this numerical method has some nice features: it is very versatile and most notably, it can be parallellized efficiently. This makes for an interesting numerical tool for the simulation of quantum relativistic dynamical phenomena such as the electron dynamics in very high intensity lasers. Moreover, this numerical scheme can be implemented on a digital quantum computer due to its simple structure: the operator splitting is a sequence of streaming operators followed by rotations in spinor space. This structure is actually reminiscent of quantum walks, which can be implemented efficiently on quantum computers. We determine the resource requirements of the resulting quantum algorithm and show that under some conditions, it has an exponential speedup over the classical algorithm. Finally, an explicit decomposition of this algorithm into elementary gates from a universal set is carried out using the software Quipper. It is shown that a proof-of-principle calculation may be possible with actual quantum technologies.