Stairway Codes: Floquetifying Bivariate Bicycle Codes and Beyond

Abstract: Floquet codes define fault-tolerant protocols through periodic measurement sequences that drive a dynamically evolving stabilizer group. They provide a natural framework for hardware supporting two-qubit parity measurements but no unitary entangling gates. However, few known constructions achieve both high encoding rates and high thresholds. We close this gap by introducing Stairway codes, a family of high-rate Floquet protocols obtained by Floquetifying Abelian two-block group algebra codes, a class that includes the bivariate bicycle codes. By representing the static code as a foliated ZX-calculus network within a $(w{-}1)$-dimensional space-time lattice and rotating the time axis, we decompose its weight-$w$ stabilizers into a periodic sequence of pairwise measurements. This reduces the design of new codes within this family to the selection of favorable periodic boundary conditions. We identify instances with competitive parameters, analyze their distance under circuit-level noise, and demonstrate logical error rates surpassing those of other Floquet codes at comparable encoding rates. Remarkably, our construction requires fewer than 300 physical qubits to match the distance and encoding rate of semi-hyperbolic Floquet codes that use over 1300 qubits.