  We introduce a quantum algorithm for simulating the time-dependent Dirac
equation in 3+1 dimensions using discrete-time quantum walks. Thus far,
promising quantum algorithms have been proposed to simulate quantum dynamics in
non-relativistic regimes efficiently. However, only some studies have attempted
to simulate relativistic dynamics due to its theoretical and computational
difficulty. By leveraging the convergence of discrete-time quantum walks to the
Dirac equation, we develop a quantum spectral method that approximates smooth
solutions with exponential convergence. This mitigates errors in implementing
potential functions and reduces the overall gate complexity that depends on
errors. We demonstrate that our approach does not require additional operations
compared to the asymptotic gate complexity of non-relativistic real-space
algorithms. Our findings indicate that simulating relativistic dynamics is
achievable with quantum computers and can provide insights into relativistic
quantum physics and chemistry.
