Integrability and equilibration in Goldilocks quantum cellular automata

We investigate the connection between Goldilocks quantum cellular automata (QCA) and free-fermion models. Our Goldilocks QCA are based upon a brickwork circuit layout of CCU gate operations with U a local unitary, the generalization of Toffoli gates CCX. Leveraging the free-fermion connection for a subclass of local unitaries U, experimentally-measurable correlation functions are efficiently simulated using Gaussian circuit dynamics, with 13 non-Abelian conserved charges explicitly identified. Using this integrable system as a basis, we then treat equilibration dynamics of both non-Gaussian initial conditions and more general non-integrable local unitaries. Beyond the QCA family studied in detail, we outline generalizations that expand the scope of free-fermion QCA. Our work raises several questions for the community to consider, chiefly, can we classify all kinetically-constrained Floquet circuits that are in fact free fermions? How will KAM theory apply for non-Abelian integrable and near integrable entangled quantum circuit dynamics? Our work provides a near-term practical road map to investigate free-fermion integrable and near-integrable dynamics on NISQ computers.