Programmable Anyon Mobility through Higher Order Cellular Automata

Abstract: Controlling anyon mobility is critical for robust quantum memory and understanding symmetry-enriched topological (SET) phases with subsystem symmetries (e.g., line-like, fractal, chaotic, or mixed supports). However, a unified framework for anyon mobility in SET phases with such diverse geometric patterns of symmetry supports has remained a major challenge. In this Letter, by introducing higher-order cellular automata (HOCA) -- a powerful computer science tool -- to SET physics, we establish a unified approach for complete characterization of anyon mobility induced by the complexity of subsystem symmetries. First, we design finite-depth HOCA-controlled unitary quantum circuits, yielding exactly solvable SET models with Abelian anyons and all possible locally generated subsystem symmetries. Then, we present a theorem that precisely programs all excitation mobilities (fractons, lineons, or fully mobile anyons) directly from the HOCA rule, representing the first complete characterization of anyon mobility in SET phases. As a corollary, this theorem yields symmetry-enriched fusion rules which govern mobility transmutation during fusion. Fusion rules with multiple channels are identified, exhibiting non-Abelian characteristics in Abelian anyon systems. Leveraging HOCA, this Letter opens new avenues for characterization of SET phases of matter and programmability of topological quantum codes.