  Dirac particle dynamics is encoded as a unitary path summation rule and
implemented on a qubit array, where the qubit array represents both spacetime
and the fermions contained therein. The unitary path summation rule gives a
quantum algorithm to model a many-body system of Dirac particles in a gauge
field with Lorentz invariance down to the grid scale (Planck scale)--the
lattice-based model neither suffers the Fermi-sign problem nor breaks Lorentz
invariance. Yet, for the Dirac Hamiltonian to generate the unitary evolution of
the 4-spinor field at the Planck scale, there is time dilation between the
shortest observable time near a single space point and that time measured at
long-wavelength scales. We find gravitational time dilation where the model
space around each point (with an even number of qubits) is curved like the
space around a Schwarzschild black hole.
