  This paper studies the quantum lattice Boltzmann scheme for the nonlinear
Dirac equations for Gross-Neveu model in $1+1$ dimensions. The initial data for
the scheme are assumed to be convergent in $L^2$. Then for any $T\ge 0$ the
corresponding solutions for the quantum lattice Boltzmann scheme are shown to
be convergent in $C([0,T];L^2(R^1))$ to the strong solution to the nonlinear
Dirac equations as the mesh sizes converge to zero. In the proof, at first a
Glimm type functional is introduced to establish the stability estimates for
the difference between two solutions for the corresponding quantum lattice
Boltzmann scheme, which leads to the compactness of the set of the solutions
for the quantum lattice Boltzmann scheme. Finally, the limit of any convergent
subsequence of the solutions for the quantum lattice Boltzmann scheme is shown
to coincide with the strong solution to a Cauchy problem for the nonlinear
Dirac equations.
