  We present an explicit time-dependent matrix product ansatz (tMPA) which
describes the time-evolution of any local observable in an interacting and
deterministic lattice gas, specifically for the rule 54 reversible cellular
automaton of [Bobenko et al., Commun. Math. Phys. 158, 127 (1993)]. Our
construction is based on an explicit solution of real-space real-time inverse
scattering problem. We consider two applications of this tMPA. Firstly, we
provide the first exact and explicit computation of the dynamic structure
factor in an interacting deterministic model, and secondly, we solve the
extremal case of the inhomogeneous quench problem, where a semi-infinite
lattice in the maximum entropy state is joined with an empty semi-infinite
lattice. Both of these exact results rigorously demonstrate a coexistence of
ballistic and diffusive transport behaviour in the model, as expected for
normal fluids.
