Floquet Codes from Coupled Spin Chains

Abstract

We propose a novel construction of the Floquet 3D toric code and Floquet $X$-cube code through the coupling of spin chains. This approach not only recovers the coupling layer construction on foliated lattices in three dimensions but also avoids the complexity of coupling layers in higher dimensions, offering a more localized and easily generalizable framework. Our method extends the Floquet 3D toric code to a broader class of lattices, aligning with its topological phase properties. Furthermore, we generalize the Floquet $X$-cube model to arbitrary manifolds, provided the lattice is locally cubic, consistent with its Fractonic phases. We also introduce a unified error-correction paradigm for Floquet codes by defining a subgroup, the Steady Stabilizer Group (SSG), of the Instantaneous Stabilizer Group (ISG), emphasizing that not all terms in the ISG contribute to error correction, but only those terms that can be referred to at least twice before being removed from the ISG. We show that correctable Floquet codes naturally require the SSG to form a classical error-correcting code, and we present a simple 2-step Bacon-Shor Floquet code as an example, where SSG forms instantaneous repetition codes. Finally, our construction intrinsically supports the extension to $n$-dimensional Floquet $(n,1)$ toric codes and generalized $n$-dimensional Floquet $X$-cube codes.