  The Thirring Quantum Cellular Automaton (QCA) describes the discrete time
dynamics of local fermionic modes that evolve according to one step of the
Dirac cellular automaton followed by the most general on-site number-preserving
interaction, and serves as the QCA counterpart of the Thirring model in quantum
field theory. In this work, we develop perturbative techniques for the QCA
path-sum approach, expanding both the number of interaction vertices and the
mass parameter of the Thirring QCA. By classifying paths within the regimes of
very light and very heavy particles, we computed the transition matrices in the
two- and three-particle sectors to the first few orders. Our investigation into
the properties of the Thirring QCA, addressing the combinatorial complexity of
the problem, yielded some useful results applicable to the many-particle sector
of any on-site number-preserving interactions in one spatial dimension.
