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

Mohammad Rowshan

TL;DR

The paper addresses decoding under dephasing-biased noise by introducing strip-symmetric biased codes, a unifying framework that explains the strip-like organization of Z-syndromes in static and Floquet codes. It formalises strip-local Z faults and per-strip $\mathbb{Z}_2$ 1-form symmetries, showing that the Z-detector incidence matrix $H_Z$ becomes block-diagonal and the maximum-likelihood decoding factorises across strips, yielding substantial decoding-complexity savings. The framework unifies known bias-tailored examples such as the XZZX surface code, domain-wall color code, and the X$^3$Z$^3$ Floquet code, and extends to Floquet codes via domain-wise Clifford deformations. Numerical benchmarks on physical codes and synthetic strip shadows confirm the predicted factorisation and establish a code-capacity Z-threshold around $p_{\mathrm{th}} = frac{1}{2}$. This design-oriented perspective provides a general toolset for creating new bias-tailored Floquet codes with tractable decoding and robust biased-performance.

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.

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

TL;DR

The paper addresses decoding under dephasing-biased noise by introducing strip-symmetric biased codes, a unifying framework that explains the strip-like organization of Z-syndromes in static and Floquet codes. It formalises strip-local Z faults and per-strip 1-form symmetries, showing that the Z-detector incidence matrix becomes block-diagonal and the maximum-likelihood decoding factorises across strips, yielding substantial decoding-complexity savings. The framework unifies known bias-tailored examples such as the XZZX surface code, domain-wall color code, and the XZ Floquet code, and extends to Floquet codes via domain-wise Clifford deformations. Numerical benchmarks on physical codes and synthetic strip shadows confirm the predicted factorisation and establish a code-capacity Z-threshold around . This design-oriented perspective provides a general toolset for creating new bias-tailored Floquet codes with tractable decoding and robust biased-performance.

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 syndromes decouple into one-dimensional repetition-like chains; the 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 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 decoding factorizes across strips, yielding complexity savings for matching-based decoders. We characterize strip symmetry via per-strip stabilizer products, viewed as a 1-form symmetry, place XZZX, the domain wall color code, and 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.
Paper Structure (16 sections, 8 theorems, 20 equations, 3 figures, 1 table)

This paper contains 16 sections, 8 theorems, 20 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Static stabilizer and dynamical codes
  4. Biased Pauli noise
  5. Detectors and Z-detector (hyper)graphs
  6. Static strip-like biased codes
  7. Strip-Symmetric Biased Codes
  8. Strip partitions and strip symmetry
  9. Block-diagonal incidence matrix and strip hypergraph
  10. Factorised ML decoding and complexity
  11. characterisation and Floquet construction
  12. Per-strip $\mathbb{Z}_2$ 1-form symmetries
  13. Static and Floquet examples
  14. Strip-symmetric Floquet codes via domain-wise Cliffords
  15. Strip Benchmarks and Numerics
  16. ...and 1 more sections

Key Result

Lemma 1

Let a code be strip-symmetric with strips $\{(S_j,D_j)\}_{j=1}^m$. Order detectors so that the rows of $H_Z$ are grouped as $(D_1,\dots,D_m)$, and group faults by strip. Then $H_Z$ is permuted into block-diagonal form where $H_j$ is the restriction of $H_Z$ to detectors in $D_j$ and faults whose flipped-detector sets lie in $D_j$.

Figures (3)

  • Figure 1: Example Z-detector hypergraph and incidence matrix $H_Z$: detectors $d_1,\dots,d_5$ are vertices, and each fault $f_\ell$ is a (hyper)edge with support $\partial(f_\ell)$.
  • Figure 2: Strip structure in synthetic benchmarks. (a) $\mathrm{DSR}(L)$: diagonal strips with nearest-neighbour ZZ repetition chains. (b) $\mathrm{CSR}(L)$: columns are vertical ZZ repetition strips. (c) $\mathrm{HCSR}(L)$: every second column active, mimicking unshaded X$^3$Z$^3$ domains and halving detector density at fixed strip length.
  • Figure 3: Logical Z error rate $P_L(p)$ vs. physical Z error rate $p$.

Theorems & Definitions (15)

  • Definition 1: Z-detector hypergraph and incidence matrix
  • Definition 2: Strip partition
  • Definition 3: Strip-local Z faults
  • Definition 4: Strip-symmetric biased code
  • Lemma 1: Block-diagonal $H_Z$
  • Corollary 1: Strip-wise hypergraph decomposition
  • Theorem 1: Factorised ML decoding under pure Z noise
  • Proposition 1: Complexity gain for superlinear decoders
  • Definition 5: Per-strip $\mathbb{Z}_2$ 1-form symmetry
  • Theorem 2: Strip symmetry and 1-form symmetries
  • ...and 5 more