⚛️ quantum physics

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

This paper demonstrates that while the standard ZX surface code's performance varies significantly with circuit-level biased noise and crosstalk, a modified ZY surface code maintains a stable Y-memory threshold across bias values, highlighting the need for decoders capable of handling correlated syndrome information to fully leverage tailored stabilizer structures.

Original authors: Pritesh Thakur, Jean-François Van Huele

Published 2026-09-11

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Pritesh Thakur, Jean-François Van Huele

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Quantum computers promise to solve problems that would take today's machines thousands of years, but they are incredibly fragile. The tiny particles that store information, called qubits, are easily disturbed by their environment, causing them to lose their data in a flash. To build a useful machine, scientists must build a shield around these qubits using a method called quantum error correction. This technique uses many physical qubits to create a single, robust "logical" qubit that can survive mistakes. The most promising shield for this job is called the surface code, which arranges qubits in a grid and constantly checks their status to catch errors before they spread. However, real-world hardware is not perfect; the errors it makes are rarely random. Often, one type of mistake happens far more frequently than others, much like a car tire that is more likely to lose air than to blow out. Understanding how to build error-correcting codes that are specifically tuned to handle these common, biased mistakes is a major goal for the field.

Researchers Pritesh Thakur and Jean-François Van Huele set out to test a new design for this protective shield, known as the ZY surface code. In the standard version of this code, the system uses two types of checks to monitor the qubits: one type looks for bit-flip errors, and the other looks for phase-flip errors. The researchers wondered what would happen if they swapped one of these checks for a different kind, creating a code that was better suited to handle a specific, dominant type of error. They simulated this new design on a computer, subjecting it to a noisy environment where one type of error was much more likely to occur than the others. They also added a second, more complex layer of noise that mimics the way quantum gates can accidentally interfere with each other, a problem known as crosstalk. Their goal was to see if this new arrangement could survive longer and more reliably than the standard design when faced with these realistic, messy conditions.

The team ran thousands of simulations using a practical decoding method that a real computer could actually use, rather than a theoretical one that is too slow for real-time use. They tested their new code against the standard version across a wide range of error rates and different ways of arranging the checks. They found that the standard code behaves in a predictable way: as the bias toward one type of error increases, the code becomes much better at protecting against that specific error, but its ability to protect against the other type of error drops and then levels off. The new ZY code, however, showed a different pattern. While it did not become dramatically better at protecting against the dominant error as the bias increased, it also did not lose its ability to protect against the other type of error. Instead, its performance remained steady and consistent across all levels of bias.

Crucially, the researchers discovered that the new code did not offer a clear advantage in terms of the maximum error rate it could tolerate before failing. While the new design provided more information about the errors occurring in the system, the practical decoder they used could not fully utilize this extra information to improve the overall threshold. The threshold, which is the point below which the code can successfully correct errors, remained statistically the same for the new code as it was for the old one. This suggests that simply changing the structure of the code is not enough; the software that reads the error signals must also be upgraded to make sense of the new, more complex patterns. The study also revealed that the order in which the checks are performed matters significantly. Changing the sequence of operations could shift the code's strength from protecting one type of information to another, allowing engineers to tune the system for specific needs, but it did not uniformly improve performance across the board.

When the researchers added the crosstalk noise, representing the interference between quantum gates, the results shifted slightly. The new code became slightly more vulnerable to errors, particularly in protecting against the less common type of mistake. This indicates that while the new design is robust, it is not immune to the correlated errors that plague real hardware. The study concludes that while tailoring the code to the noise is a promising idea, the current tools for reading the error signals are not sophisticated enough to fully exploit the benefits of these new structures. To truly unlock the potential of these specialized codes, the field needs to develop faster, smarter decoders that can reason about the complex, interconnected nature of the errors, rather than just matching them up in pairs. Until then, the standard code remains a reliable, if not perfectly optimized, choice for protecting quantum information.

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

Problem Statement
Quantum error correction (QEC) is essential for fault-tolerant quantum computing, particularly given the prevalence of biased noise in physical hardware (e.g., superconducting qubits and trapped ions where dephasing or bit-flip errors dominate). While previous research by Tuckett et al. (2018) demonstrated that tailoring surface code stabilizers (specifically replacing Z-type with Y-type stabilizers) significantly improves thresholds under code-capacity noise using optimal tensor-network decoders, it remains an open question whether these benefits persist under circuit-level noise with practical, real-time decoders. Circuit-level noise introduces faults in every gate, measurement, and reset, and requires decoders capable of handling three-dimensional (space-time) syndrome data. Furthermore, realistic hardware often exhibits correlated errors, such as gate-based crosstalk, which may degrade performance. This work investigates whether the structural advantages of tailored stabilizers translate to circuit-level thresholds using practical matching decoders and whether crosstalk noise impacts these tailored codes differently than standard codes.

Methodology
The authors constructed and simulated a ZY surface code, where the standard X-type stabilizers are replaced with Y-type stabilizers, while retaining Z-type stabilizers. This construction was compared against the standard ZX surface code. The study utilized the following parameters and tools:

Noise Models: Two circuit-level noise models were implemented: (1) Pauli-X biased noise (where bit-flip errors dominate), characterized by a bias parameter $\eta$ ranging from unbiased ( $\eta=0.5$ ) to pure X-bias ( $\eta=\infty$ ); and (2) a combined model adding gate-based XX crosstalk noise ( $p_\times = 5 \times 10^{-4}$ ) to simulate residual interactions in superconducting and trapped-ion systems.
Code Variants: Both rotated and unrotated surface code geometries were simulated with code distances $d \in \{5, \dots, 17\}$ .
CNOT Scheduling: Two CNOT ordering schedules were tested for the ZX code: "X-aligned" (10231203) and "Z-aligned" (01320312). For the ZY code, only the X-aligned schedule was used.
Decoding: The PyMatching v2.3.1 decoder (Minimum-Weight Perfect Matching with correlated matching support) was used for all simulations.
Simulation Pipeline: Monte Carlo simulations were performed using Stim and Sinter, tracking logical error rates for X, Y, and Z memory experiments. Thresholds were extracted using critical exponent scaling fits.

Key Contributions

Extension to Circuit-Level Noise: The work extends the observation of tailored stabilizer benefits from the code-capacity regime to the more realistic circuit-level regime, incorporating full gate, measurement, and reset faults.
ZY Surface Code Construction: The authors formally define and simulate a ZY surface code under X-biased noise, providing a direct comparison with the standard ZX code under identical circuit-level conditions.
Crosstalk Analysis: The study quantifies the impact of gate-based XX crosstalk on both standard and tailored surface codes, distinguishing between effects on different logical memories (X/Y vs. Z).
Decoder Limitations: The paper highlights the limitations of matching-based decoders (specifically PyMatching) in exploiting the additional syndrome information provided by tailored stabilizer structures.

Results

ZX Surface Code Behavior:
- Under X-biased noise, the X-memory threshold ( $p^X_{th}$ ) increases monotonically with bias $\eta$ , eventually exceeding the simulation window for pure X-bias. Conversely, the Z-memory threshold ( $p^Z_{th}$ ) decreases and saturates at a low value.
- Adding XX crosstalk significantly reduces the Z-memory threshold (beyond fitting uncertainty) but leaves the X-memory threshold largely unaffected.
- CNOT Ordering: The choice of CNOT schedule redistributes performance. The Z-aligned schedule improves the Z-memory threshold (especially in unrotated codes) at the cost of slightly reducing the X-memory threshold at high bias.
ZY Surface Code Behavior:
- Y-Memory Stability: Unlike the ZX code's X-memory, the Y-memory threshold ( $p^Y_{th}$ ) of the ZY code remains nearly constant across all bias values, showing no significant improvement with increased bias.
- Z-Memory Equivalence: The Z-memory thresholds of the ZY and ZX codes are statistically indistinguishable across all bias values.
- Crosstalk Impact: Adding crosstalk reduces both Y and Z memory thresholds for the ZY code. The reduction in Y-memory is significant across all bias values.
Decoder Performance: The ZY code provides twice the syndrome information regarding dominant X errors compared to the ZX code. However, the PyMatching decoder fails to utilize this extra information to improve the Z-memory threshold, resulting in performance identical to the standard ZX code.

Significance and Claims
The paper claims that while tailored stabilizer structures (like the ZY code) offer theoretical advantages by doubling syndrome information for dominant errors, practical matching decoders (MWPM) cannot fully exploit this structure under circuit-level noise. The authors conclude that the additional syndrome information in the ZY code forms a hypergraph structure that standard matching algorithms cannot process effectively without joint reasoning over correlated syndromes.

Consequently, the work suggests that for matching-based decoders, modifying the code to simplify error geometry may be more effective than enriching the syndrome structure. The results indicate a clear need for decoders capable of jointly reasoning over correlated syndrome information (such as tensor-network decoders or learning-based approaches) to fully realize the potential of tailored stabilizer structures in circuit-level environments. The study also demonstrates that CNOT ordering is a tunable parameter that can be used to prioritize the protection of specific logical memories (X/Y vs. Z) depending on application requirements.

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