Strip-Symmetric Quantum Codes for Biased Noise: Z-Decoupling in Stabilizer and Floquet Codes

TL;DR

This paper introduces strip-symmetric biased quantum codes that simplify decoding by confining errors to strips, enabling efficient fault-tolerant quantum computation under biased noise conditions.

Contribution

The paper defines a new class of strip-symmetric codes, unifies existing bias-tailored codes within this framework, and provides design tools for creating new Floquet codes with improved decoding efficiency.

Findings

Strip symmetry leads to block-diagonal detector matrices.

Decoding complexity is reduced via factorization across strips.

Framework applies to XZZX, domain wall color code, and $X^3Z^3$ codes.

Abstract

Bias-tailored codes such as the XZZX surface code and the domain wall color code achieve high dephasing-biased thresholds because, in the infinite-bias limit, their $Z$ syndromes decouple into one-dimensional repetition-like chains; the $X^3Z^3$ Floquet code shows an analogous strip-wise structure for detector events in spacetime. We capture this common mechanism by defining strip-symmetric biased codes, a class of static stabilizer and dynamical (Floquet) codes for which, under pure dephasing and perfect measurements, each elementary $Z$ fault is confined to a strip and the Z-detector--fault incidence matrix is block diagonal. For such codes the Z-detector hypergraph decomposes into independent strip components and maximum-likelihood $Z$ decoding factorizes across strips, yielding complexity savings for matching-based decoders. We characterize strip symmetry via per-strip stabilizer products, viewed as a $\mathbb{Z}_2$ 1-form symmetry, place XZZX, the domain wall color code, and $X^3Z^3$ in this framework, and introduce synthetic strip-symmetric detector models and domain-wise Clifford constructions that serve as design tools for new bias-tailored Floquet codes.