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arXiv:2610.06059

Comparing thresholds of XZZX and rotated surface codes under spatio-temporal correlated noise

Dhruv Koul

quant-phSubmitted Oct 5, 2026• Announced Oct 6, 2026• 0 citations
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Abstract

While independent and identically distributed noise is typically assumed in quantum computing research, complex error models such as correlated and spatio-temporal errors remain underexplored. In this paper, the performance of $\mathrm{XZZX}$ and rotated surface codes is evaluated under these complex noise models. Standard graph-based decoders struggle with the hyperedges generated by correlated errors on bias-tailored codes. To address this, we employ the Minimum Weight Parity Factor (MWPF) decoder, which natively processes syndrome hypergraphs without artificial edge decomposition. Our simulations show that under strong Z-biased circuit-level correlated noise ($η=100$), the threshold of the rotated code degrades to $\approx 0.36\%$, while the $\mathrm{XZZX}$ code drops to $\approx 0.70\%$. Furthermore, a distinct non-monotonic threshold dependence is observed for the $\mathrm{XZZX}$ code across different bias ratios. While the $\mathrm{XZZX}$ code effectively absorbs high-density Z-type hyperedges at extreme biases, it exhibits a threshold minimum at moderate biases ($η\approx 10$) because transverse X and Y errors actively bridge the diagonal strings, breaking the code's one-dimensional decoupling symmetry. These findings demonstrate that utilizing hypergraph-aware decoders is critical, as the performance of bias-tailored codes is highly sensitive to the geometric structure of spatio-temporal correlated noise.

Comments
11 pages, 8 plots, simulation codes available on GitHub

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While independent and identically distributed noise is typically assumed in quantum computing research, complex error models such as correlated and spatio-temporal errors remain underexplored. In this paper, the performance of XZZX\mathrm{XZZX}XZZX and rotated surface codes is evaluated under these complex noise models. Standard graph-based decoders struggle with the hyperedges generated by correlated errors on bias-tailored codes. To address this, we employ the Minimum Weight Parity Factor (MWPF) decoder, which natively processes syndrome hypergraphs without artificial edge decomposition. Our simulations show that under strong Z-biased circuit-level correlated noise (η=100η=100η=100), the threshold of the rotated code degrades to ≈0.36%\approx 0.36\%≈0.36%, while the XZZX\mathrm{XZZX}XZZX code drops to ≈0.70%\approx 0.70\%≈0.70%. Furthermore, a distinct non-monotonic threshold dependence is observed for the XZZX\mathrm{XZZX}XZZX code across different bias ratios. While the XZZX\mathrm{XZZX}XZZX code effectively absorbs high-density Z-type hyperedges at extreme biases, it exhibits a threshold minimum at moderate biases (η≈10η\approx 10η≈10) because transverse X and Y errors actively bridge the diagonal strings, breaking the code's one-dimensional decoupling symmetry. These findings demonstrate that utilizing hypergraph-aware decoders is critical, as the performance of bias-tailored codes is highly sensitive to the geometric structure of spatio-temporal correlated noise.
Studying the threshold behavior of surface codes under biased noise models is an active area of research. Previous work Tuckett et al. (2018), using an optimal tensor-network decoder, demonstrated that replacing ZZZ-type stabilizers with YYY-type stabilizers significantly improves the surface code threshold under code-capacity level dephasing noise. In this work, we construct and study a ZYZYZY surface code by replacing the XXX-type stabilizers with YYY-type stabilizers. We compare it with the standard ZXZXZX surface code under circuit-level Pauli-XXX biased noise, with and without an additional gate-based XXXXXX crosstalk noise. We find that for the ZXZXZX surface code, the XXX-memory threshold increases monotonically with bias while the ZZZ-memory threshold decreases and saturates. For the ZYZYZY surface code, the YYY-memory threshold is nearly constant across all bias values. The ZZZ-memory thresholds of the ZXZXZX and ZYZYZY codes are consistent within the uncertainty. Adding XXXXXX crosstalk reduces the ZZZ-memory threshold beyond the fitting uncertainty while leaving the XXX memory threshold largely unaffected. The choice of CNOT ordering redistributes threshold performance between the two logical memories. Our work extends prior observations from code-capacity level noise to circuit-level noise. It also indicates the need for decoders capable of jointly reasoning over correlated syndrome information so that tailored stabilizer structures could be fully utilized for quantum error correction.
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 ZZZ syndromes decouple into one-dimensional repetition-like chains; the X3Z3X^3Z^3X3Z3 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 ZZZ 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 ZZZ decoding factorizes across strips, yielding complexity savings for matching-based decoders. We characterize strip symmetry via per-strip stabilizer products, viewed as a Z2\mathbb{Z}_2Z2​ 1-form symmetry, place XZZX, the domain wall color code, and X3Z3X^3Z^3X3Z3 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.
Quantum error correction (QEC) is often implemented on hardware that experiences biased noise, where dephasing errors occur more frequently than other errors. This has motivated many recent efforts to develop bias-tailored QEC codes, such as the Clifford-deformed compass codes: a family of codes that achieve high thresholds under biased dephasing noise. We perform circuit-level simulations of the Clifford-deformed elongated compass codes under a biased noise model and evaluate code thresholds using standard minimum weight perfect matching (MWPM) and correlated MWPM. We find that correlated decoding enhances thresholds for all noise biases relative to standard MWPM under circuit-level noise. Our results demonstrate that correlated decoding leads to a higher relative gain in thresholds compared to standard MWPM when applied to codes with asymmetric stabilizers under biased noise.
The minimum-weight perfect matching (MWPM) decoder is a standard decoding strategy for surface codes, but its performance can degrade considerably under biased noise. In this work, a modified family of surface codes, termed the XYZ planar code, is introduced, and MWPM is extended to posterior MWPM (pMWPM), which exploits the information provided by mixed stabilizers with almost no increase in decoding complexity. The XYZ--pMWPM and XZZX code--decoder combinations are compared under homogeneous independent and identically distributed (IID) Pauli noise and processor-derived independent nonidentically distributed (INID) noise. Under homogeneous biased Pauli noise, pMWPM substantially increases the threshold of the XYZ planar code relative to conventional MWPM. At finite code distances, particularly for small codes under strong bias, XYZ--pMWPM achieves logical error rates comparable to or lower than those of XZZX. Under INID noise, the pseudothreshold ordering observed under the corresponding homogeneous IID references is not always preserved; in some hardware--distance settings, XYZ--pMWPM achieves pseudothresholds comparable to or higher than those of XZZX. Within XZZX, qubit-specific weighting consistently yields higher INID pseudothresholds than average-rate weighting for the hardware instances considered. It is further found that the IID--INID pseudothreshold shift depends strongly on the physical-error scalarization and the construction of the homogeneous IID reference. These results show that pMWPM efficiently exploits the mixed-stabilizer structure of the XYZ planar code and provides competitive finite-distance decoding under biased and spatially nonuniform noise.
We find that the substantial advantages predicted for bias-tailored quantum error correction (QEC) under code-capacity noise are strongly reduced once realistic syndrome extraction and circuit-level noise models are considered. We start by comparing XZZX codes to rectangular surface codes with a bias-dependent optimised anisotropy. Although code-capacity simulations predict an advantage of rectangular surface codes in the limit of high noise bias, this actually disappears under circuit-level noise, making the XZZX codes the preferred and simplest choice even for platforms that allow for a flexible variation of the code layout adapted to changes in noise calibration. Our results identify bias degradation during syndrome extraction under circuit-level noise as the central limitation of biased-tailored QEC. To partially mitigate this effect, we introduce a bias-filtering CNOT gadget that temporarily encodes the ancillary target qubit during syndrome extraction in a repetition code and, upon measurement and feed forward, manages to reduce the bias degradation. In a regime of high-bias and low-idle errors, this bias-filtering gadget yields a few-percent relative improvement of the XZZX code error threshold, demonstrating that lightweight bias-filtering strategies can recover part of the lost bias-tailoring advantage for realistic circuit-level noise.
Noise in quantum computing is countered with quantum error correction. Achieving optimal performance will require tailoring codes and decoding algorithms to account for features of realistic noise, such as the common situation where the noise is biased towards dephasing. Here we introduce an efficient high-threshold decoder for a noise-tailored surface code based on minimum-weight perfect matching. The decoder exploits the symmetries of its syndrome under the action of biased noise and generalizes to the fault-tolerant regime where measurements are unreliable. Using this decoder, we obtain fault-tolerant thresholds in excess of 6% for a phenomenological noise model in the limit where dephasing dominates. These gains persist even for modest noise biases: we find a threshold of ∼5% in an experimentally relevant regime where dephasing errors occur at a rate 100 times greater than bit-flip errors.