Clifford-Deformed Compass Codes

1. Problem Statement

Quantum error correction (QEC) is essential for fault-tolerant quantum computing, but standard noise models (like depolarizing noise) often fail to capture the specific characteristics of real-world hardware. Many quantum architectures (e.g., superconducting cat qubits, trapped ions) exhibit biased noise, where dephasing ($Z$) errors occur much more frequently than bit-flip ($X$) or bit-phase-flip ($Y$) errors.

While existing codes like the XZZX surface code and elongated compass codes have shown promise under biased noise, they have limitations:

• Elongated Compass Codes: These codes optimize performance for specific bias levels by increasing the number of weight-2 $X$ stabilizers. However, their performance peaks at a specific "optimal bias" ($\eta_{opt}$) and degrades as the bias increases further due to an asymmetry in $X$ and $Z$ decoding capabilities.
• XZZX Surface Code: While robust across a wide range of biases, it may not fully exploit the structural advantages of elongated compass codes for specific high-bias regimes.

The authors aim to design QEC codes that maintain the structural benefits of elongated compass codes while introducing symmetries to improve thresholds and logical error rates across a broader spectrum of high-bias noise models.

2. Methodology

Noise Model

The authors utilize a single-qubit Pauli noise channel where errors occur with probabilities $p_x, p_y, p_z$. The noise is biased towards dephasing ($Z$ errors), quantified by the bias parameter $\eta = p_z / (p_x + p_y)$. They evaluate codes for $\eta$ ranging from 0.5 (unbiased/depolarizing) to 100.

• Code Capacity Model: Assumes perfect syndrome extraction; errors occur only on data qubits.
• Phenomenological Model: Includes measurement errors. Crucially, measurement error rates are weighted by the stabilizer weight ($w$) to account for the deeper circuits required for high-weight stabilizers in elongated compass codes.

Code Construction: Clifford Deformations

The core methodology involves applying Clifford deformations to elongated compass codes. A Clifford deformation modifies stabilizers via single-qubit Clifford operators (specifically Hadamard gates) without changing the code's distance or support qubits.

• Target: Elongated compass codes with elongation parameter $\ell$ (where $\ell=2$ is the rotated surface code).
• Strategy: The authors introduce two specific deformations designed to preserve the weight-2 $X$ stabilizers (which are crucial for detecting $Z$ errors) while introducing symmetry to restrict defect propagation:
  1. $XZZX_{\square}$ Deformation: Applies Hadamard gates to the top-right and bottom-left qubits of weight-4 $X$ stabilizers. This transforms them into $XZZX$ form.
  2. $ZXXZ_{\square}$ Deformation: Applies Hadamard gates to the top-left and bottom-right qubits of weight-4 $X$ stabilizers. This affects only the top and bottom rows of the code.

Decoding

The authors use the Minimum-Weight Perfect Matching (MWPM) decoder.

• Since the deformed codes are no longer CSS (Calderbank-Shor-Steane) codes, $X$ and $Z$ syndromes cannot be decoded independently in the standard way.
• Solution: The authors map the decoding problem of the deformed code under biased noise to an equivalent decoding problem of the undeformed CSS code under an inhomogeneous noise model. By adjusting edge weights in the decoder graph based on which qubits underwent Hadamard transformations, they can use standard CSS decoder graphs to decode the deformed syndromes.

3. Key Contributions

1. Novel Code Families: The introduction of $XZZX_{\square}$ and $ZXXZ_{\square}$ deformed elongated compass codes. These codes preserve the high-weight $X$ stabilizers of the original compass codes while introducing symmetries that restrict the spread of defects.
2. Decoder Graph Analysis: A detailed analysis of how these deformations alter the decoder graphs.
   • The $XZZX_{\square}$ deformation partitions the low-weight graph into disjoint regions, restricting high-rate error spread but creating high-degree vertices in the high-weight graph.
   • The $ZXXZ_{\square}$ deformation creates disjoint strings in the low-weight graph and partitions the high-weight graph, offering a more balanced structure for moderate biases.
3. Performance Benchmarking: Comprehensive Monte Carlo simulations comparing these new codes against standard CSS elongated compass codes and the XZZX surface code under both code capacity and phenomenological noise models.

4. Results

Code Capacity Noise (Ideal Syndrome Extraction)

• Threshold Trends:
  • Standard CSS elongated compass codes reach a maximum threshold at a specific optimal bias ($\eta_{opt}$) and flatten out as bias increases.
  • $XZZX_{\square}$ codes show increasing thresholds with bias, but the advantage of higher elongation ($\ell$) diminishes at very high biases.
  • $ZXXZ_{\square}$ codes exhibit the most significant improvement. Their thresholds increase rapidly with bias and surpass the XZZX surface code thresholds for moderate to high biases ($10 < \eta \le 100$).
• Logical Error Rates:
  • At physical error rates $p=0.05$ and $p=0.10$, the $ZXXZ_{\square}$ deformed codes achieve the lowest logical error rates among all tested codes for biases $\eta \ge 10$.
  • Notably, at certain biases, the $ZXXZ_{\square}$ codes outperform the XZZX surface code in logical error rate suppression.

Phenomenological Noise (Realistic Syndrome Extraction)

• When accounting for measurement errors weighted by stabilizer size, the performance landscape shifts.
• The $ZXXZ_{\square}$ codes still show increasing thresholds with bias.
• However, the XZZX surface code achieves the highest absolute thresholds in this regime. The increased stabilizer weight of the elongated compass codes introduces more measurement errors, negating the theoretical threshold gains seen in the code capacity model.
• Despite this, the deformed codes still successfully achieve the goal of having thresholds that increase with bias, unlike the standard CSS codes.

5. Significance and Conclusion

• Optimization for Biased Hardware: The work demonstrates that by tailoring Clifford deformations to specific hardware noise biases, one can construct codes that outperform the current state-of-the-art (XZZX surface code) in the code capacity limit.
• Trade-off Management: The study highlights the trade-off between the theoretical benefits of elongated codes (better $Z$ error detection) and the practical costs of high-weight stabilizers (increased measurement errors).
• Future Directions: The authors suggest that while the phenomenological results favor the XZZX surface code, the deformed compass codes could be superior if implemented with circuit-level noise models that utilize bias-preserving gates and optimized syndrome extraction schedules.
• Broader Impact: The methodology of applying Clifford deformations to non-surface codes (like compass codes) to engineer specific decoder graph structures offers a general framework for designing QEC codes for diverse quantum architectures, including measurement-based and fusion-based quantum computing.

In summary, this paper successfully extends the utility of elongated compass codes by applying targeted Clifford deformations, resulting in the $ZXXZ_{\square}$ code, which offers superior logical error rates and thresholds over the XZZX surface code under high-bias conditions in the absence of circuit-level overhead.