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arXiv:2609.10876

Threshold Behavior of ZX and ZY Surface Codes Under Circuit-Level Biased and Crosstalk Noise

Pritesh Thakur, Jean-François Van Huele

quant-phSubmitted Sep 9, 2026• Announced Sep 11, 2026• 0 citations
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Abstract

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 $Z$-type stabilizers with $Y$-type stabilizers significantly improves the surface code threshold under code-capacity level dephasing noise. In this work, we construct and study a $ZY$ surface code by replacing the $X$-type stabilizers with $Y$-type stabilizers. We compare it with the standard $ZX$ surface code under circuit-level Pauli-$X$ biased noise, with and without an additional gate-based $XX$ crosstalk noise. We find that for the $ZX$ surface code, the $X$-memory threshold increases monotonically with bias while the $Z$-memory threshold decreases and saturates. For the $ZY$ surface code, the $Y$-memory threshold is nearly constant across all bias values. The $Z$-memory thresholds of the $ZX$ and $ZY$ codes are consistent within the uncertainty. Adding $XX$ crosstalk reduces the $Z$-memory threshold beyond the fitting uncertainty while leaving the $X$ 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.

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Applying single-qubit Clifford unitaries to a Pauli stabilizer code produces a Clifford-deformed variant whose stabilizers remain Pauli operators, but with locally rotated Pauli axes. Such deformations provide a simple way to tailor a fixed code to anisotropic noise, and have enabled unusually high thresholds under strongly biased dephasing. In this work, we discuss zero-rate quantum low-density parity-check (LDPC) codes, for which there exist Clifford-deformed variants where the number of biased logical operators scales slower than the distance, or there exists a basis of logical operators whose overlap satisfies certain scaling conditions; in this case, the code-capacity threshold for the Clifford-deformed variant under i.i.d. pure dephasing noise approaches 50%. This property provably explains previously known code examples with 50% biased noise thresholds, such as XY surface code, XZZX surface code, color code, as well as some 3D Clifford-deformed codes. As a concrete new example, we study Clifford deformations of the tile codes of Ref. [1]. Similar to the phase diagram of 50% thresholds for random Clifford deformations of the surface code in Ref. [2], we find a similar phase diagram for the tile codes. We also construct several translationally invariant deformations of the tile code with 50% thresholds, and present numerical evidence for improved performance at finite bias and under circuit-level noise. In the circuit-level setting, performance is governed by the residual bias after a full syndrome-extraction cycle, linking our simulations to phenomenological models commonly used to study Clifford-deformed codes. We estimate this residual bias for different qubit platforms by modeling microscopic implementations of tile-code syndrome extraction.

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Leveraging biased noise for more efficient quantum error correction at the circuit level with two-level qubits

Josu Etxezarreta Martínez, Paul Schnabl, Javier Oliva del Moral +3 more

Tailoring quantum-error-correction codes (QECCs) to biased noise has demonstrated significant benefits. However, most of the prior research on this topic has focused on code-capacity noise models. Furthermore, a no-go theorem prevents the construction of controlled- () gates for two-level qubits in a bias-preserving manner which may, in principle, imply that noise bias cannot be leveraged in such systems. In this work, we show that a residual bias up to η ∼ 5 can be maintained in gates under certain conditions. Moreover, we employ controlled- () gates in syndrome-extraction circuits and show how to natively implement these in a bias-preserving manner for a broad class of qubit platforms. This motivates the introduction of what we call a (HBD) circuit-level noise model that captures these features. We numerically study the performance of the X Z Z X surface code and observe that bias-preserving gates are critical for leveraging biased noise. Accounting for the residual bias present in the gates, we observe an increase in the code threshold up to a 1.27 % physical error rate, representing a 90 % improvement. Additionally, we find that the required qubit footprint can be reduced by up to 75 % at relevant physical error rates.

arXiv:2505.17718•Physical Review Applied•Dec 16, 2025•1 citation
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.
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.
We show that a simple modification of the surface code can exhibit an enormous gain in the error correction threshold for a noise model in which Pauli Z errors occur more frequently than X or Y errors. Such biased noise, where dephasing dominates, is ubiquitous in many quantum architectures. In the limit of pure dephasing noise we find a threshold of 43.7(1)% using a tensor network decoder proposed by Bravyi, Suchara, and Vargo. The threshold remains surprisingly large in the regime of realistic noise bias ratios, for example 28.2(2)% at a bias of 10. The performance is, in fact, at or near the hashing bound for all values of the bias. The modified surface code still uses only weight-4 stabilizers on a square lattice, but merely requires measuring products of Y instead of Z around the faces, as this doubles the number of useful syndrome bits associated with the dominant Z errors. Our results demonstrate that large efficiency gains can be found by appropriately tailoring codes and decoders to realistic noise models, even under the locality constraints of topological codes.
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.
For quantum computers to reach their full potential will require error correction. We study the surface code, one of the most promising quantum error correcting codes, in the context of predominantly dephasing (Z-biased) noise, as found in many quantum architectures. We find that the surface code is highly resilient to Y-biased noise, and tailor it to Z-biased noise, whilst retaining its practical features. We demonstrate ultrahigh thresholds for the tailored surface code: ~39% with a realistic bias of  = 100, and ~50% with pure Z noise, far exceeding known thresholds for the standard surface code: ~11% with pure Z noise, and ~19% with depolarizing noise. Furthermore, we provide strong evidence that the threshold of the tailored surface code tracks the hashing bound for all biases. We reveal the hidden structure of the tailored surface code with pure Z noise that is responsible for these ultrahigh thresholds. As a consequence, we prove that its threshold with pure Z noise is 50%, and we show that its distance to Z errors, and the number of failure modes, can be tuned by modifying its boundary. For codes with appropriately modified boundaries, the distance to Z errors is O(n) compared to O(n1/2) for square codes, where n is the number of physical qubits. We demonstrate that these characteristics yield a significant improvement in logical error rate with pure Z and Z-biased noise. Finally, we introduce an efficient approach to decoding that exploits code symmetries with respect to a given noise model, and extends readily to the fault-tolerant context, where measurements are unreliable. We use this approach to define a decoder for the tailored surface code with Z-biased noise. Although the decoder is suboptimal, we observe exceptionally high fault-tolerant thresholds of ~5% with bias  = 100 and exceeding 6% with pure Z noise. Our results open up many avenues of research and, recent developments in bias-preserving gates, highlight their direct relevance to experiment.
Applying single-qubit Clifford unitaries to a Pauli stabilizer code produces a Clifford-deformed variant whose stabilizers remain Pauli operators, but with locally rotated Pauli axes. Such deformations provide a simple way to tailor a fixed code to anisotropic noise, and have enabled unusually high thresholds under strongly biased dephasing. In this work, we discuss zero-rate quantum low-density parity-check (LDPC) codes, for which there exist Clifford-deformed variants where the number of biased logical operators scales slower than the distance, or there exists a basis of logical operators whose overlap satisfies certain scaling conditions; in this case, the code-capacity threshold for the Clifford-deformed variant under i.i.d. pure dephasing noise approaches 50%. This property provably explains previously known code examples with 50% biased noise thresholds, such as XY surface code, XZZX surface code, color code, as well as some 3D Clifford-deformed codes. As a concrete new example, we study Clifford deformations of the tile codes of Ref. [1]. Similar to the phase diagram of 50% thresholds for random Clifford deformations of the surface code in Ref. [2], we find a similar phase diagram for the tile codes. We also construct several translationally invariant deformations of the tile code with 50% thresholds, and present numerical evidence for improved performance at finite bias and under circuit-level noise. In the circuit-level setting, performance is governed by the residual bias after a full syndrome-extraction cycle, linking our simulations to phenomenological models commonly used to study Clifford-deformed codes. We estimate this residual bias for different qubit platforms by modeling microscopic implementations of tile-code syndrome extraction.
Tailoring quantum-error-correction codes (QECCs) to biased noise has demonstrated significant benefits. However, most of the prior research on this topic has focused on code-capacity noise models. Furthermore, a no-go theorem prevents the construction of controlled- () gates for two-level qubits in a bias-preserving manner which may, in principle, imply that noise bias cannot be leveraged in such systems. In this work, we show that a residual bias up to η ∼ 5 can be maintained in gates under certain conditions. Moreover, we employ controlled- () gates in syndrome-extraction circuits and show how to natively implement these in a bias-preserving manner for a broad class of qubit platforms. This motivates the introduction of what we call a (HBD) circuit-level noise model that captures these features. We numerically study the performance of the X Z Z X surface code and observe that bias-preserving gates are critical for leveraging biased noise. Accounting for the residual bias present in the gates, we observe an increase in the code threshold up to a 1.27 % physical error rate, representing a 90 % improvement. Additionally, we find that the required qubit footprint can be reduced by up to 75 % at relevant physical error rates.