  Quantum cellular automata (QCA) constitute space and time homogeneous
discrete models for quantum field theories (QFTs). Although QFTs are defined
without reference to particles, computations are done in terms of Feynman
diagrams, which are explicitly interpreted in terms of interacting particles.
Similarly, the easiest QCA to construct are quantum lattice gas automata
(QLGA). A natural question then is, which QCA are not QLGA? Here we construct a
non-trivial example of such a QCA; it provides a simple model in $1+1$
dimensions with no particle interpretation at the scale where the QCA dynamics
are homogeneous.
