Competing automorphisms and disordered Floquet codes

Topological order is a promising basis for quantum error correction, a key milestone towards large-scale quantum computing. Floquet codes provide a dynamical scheme for this while also exhibiting Floquet-enriched topological order (FET) where anyons periodically undergo a measurement-induced automorphism that acts uniformly in space. We study disordered Floquet codes where automorphisms have a spatiotemporally heterogeneous distribution -- the automorphisms "compete". We characterize the effect of this competition, showing how key features of the purification dynamics of mixed codestates can be inferred from anyon and automorphism properties for any Abelian topological order. This perspective can explain the protection or measurement of logical information in a dynamic automorphism (DA) code when subjected to a noise model of missing measurements. We demonstrate this using a DA color code with perturbed measurement sequences. The framework of competing automorphisms captures essential features of Floquet codes and robustness to noise, and may elucidate key mechanisms involving topological order, automorphisms, and fault-tolerance.