Quantum Cellular Automata: From Integrable Three-Site Dynamics to Solitonic Five-Site Extensions

We investigate quantum cellular automata (QCA), self-similar circuits generating rich dynamics via local rules. Discrete three-site Goldilocks QCA map into free fermions yielding exact local conserved quantities. Our analysis of integrability-breaking perturbations reveals the onset of universal chaos characterized by critical time scales scaling with system size. However, extended five-site QCA generate Quantum Entangled Breathers (QEBs)—robust dynamical features that resist equilibration for extended periods. These are a quantum analog of classical blinkers but with rich entanglement dynamics. Beyond breathers, five-site rules also support both bright and dark soliton solutions with distinct propagation characteristics. The stability of these solitonic states depends critically on specific local neighborhood rules whose existence strongly suggests underlying integrability. Our analysis reveals that local rule structure directly governs soliton existence through neighborhood interaction patterns, establishing a fundamental connection between discrete quantum dynamics and continuous field-theory phenomena.