  This work provides a relativistic, digital quantum simulation scheme for both
$2+1$ and $3+1$ dimensional quantum electrodynamics (QED), based on a discrete
spacetime formulation of theory. It takes the form of a quantum circuit,
infinitely repeating across space and time, parametrised by the discretization
step $\Delta_t=\Delta_x$. Strict causality at each step is ensured as circuit
wires coincide with the lightlike worldlines of QED; simulation time under
decoherence is optimized. The construction replays the logic that leads to the
QED Lagrangian. Namely, it starts from the Dirac quantum walk, well-known to
converge towards free relativistic fermions. It then extends the quantum walk
into a multi-particle sector quantum cellular automata in a way which respects
the fermionic anti-commutation relations and the discrete gauge invariance
symmetry. Both requirements can only be achieved at cost of introducing the
gauge field. Lastly the gauge field is given its own electromagnetic dynamics,
which can be formulated as a quantum walk at each plaquette.
