Erasure Thresholds for Hyperbolic and Semi-Hyperbolic Surface Codes

Technical Deep Dive: Erasure Thresholds for Hyperbolic and Semi-Hyperbolic Surface Codes

Overview

This work extends the analysis of erasure thresholds for quantum error correcting surface codes from planar to hyperbolic geometries. Key findings:

• Hyperbolic surface codes under circuit-level erasure noise exhibit erasure thresholds several times higher than their corresponding Pauli thresholds, matching the ratio observed for planar surface codes.
• For the ${8, 3}$ Bolza family of hyperbolic codes, the erasure threshold reaches 4.7% while the Pauli threshold is 0.72% - a ratio of 6.5x.
• Fine-grained "semi-hyperbolic" codes, which interpolate between hyperbolic and Euclidean scaling, also show the erasure advantage, with an erasure-to-Pauli ratio of 5.3x.
• These results establish that the benefits of erasure-aware decoding extend to the hyperbolic surface code architecture, which encodes a linear number of logical qubits into a quadratic number of physical qubits.

Problem & Context

• Quantum error correction (QEC) protects logical quantum information by encoding it redundantly across many physical qubits.
• QEC performance depends on the noise model - matching the decoder to the noise structure can improve thresholds.
• Erasure noise, where error locations are known, can provide significantly higher thresholds than standard Pauli noise models.
• Prior work has shown erasure thresholds 4-6x higher than Pauli thresholds for planar surface codes.
• This work extends the analysis to hyperbolic surface codes, which offer a different trade-off between encoding rate and code distance.

Methodology

• Construct 14 hyperbolic CSS surface codes and 11 "semi-hyperbolic" codes across the ${8, 3}$, ${10, 3}$, and ${12, 3}$ tessellation families.
• Simulate these codes under circuit-level erasure noise, circuit-level Pauli noise, and phenomenological noise models.
• Use weighted Union-Find decoding on the separate X and Z syndrome graphs.
• Estimate pseudothresholds and crossing-point family thresholds.
• Adopt the circuit-level erasure noise model of Chang et al., including imperfect erasure checks.

Data & Experimental Setup

• Simulate 6 ${8, 3}$ base codes, 4 ${10, 3}$ base codes, and 4 ${12, 3}$ base codes.
• Also simulate 11 semi-hyperbolic codes derived by fine-graining the ${8, 3}$ Bolza family.
• Use 2000 Pauli shots, 400 erasure shots per data point for base codes.
• Use 1000 Pauli shots, 200 erasure shots per data point for fine-grained codes.
• Compute pseudothresholds as the physical error rate where the per-cycle logical error rate equals the unencoded qubit error rate.
• Estimate family thresholds from per-observable crossing points between consecutive code sizes.

Results

Base Code Thresholds

• ${8, 3}$ base codes achieve Pauli pseudothresholds of 0.24-0.49%, increasing with code size.
• Under erasure noise, 3 ${8, 3}$ codes have measurable pseudothresholds of 3.3-3.7%; larger codes exceed the tested range.
• ${10, 3}$ base codes achieve Pauli pseudothresholds of 0.11-0.43%.
• ${12, 3}$ base codes all have distance 3 and Pauli pseudothresholds of 0.07-0.13%.

Fine-Grained Code Thresholds

• The ${8, 3}$ Bolza fine-grained family achieves Pauli pseudothresholds up to 0.63% and erasure pseudothresholds up to 4.4%.
• At $R_e = 1$, the Bolza family threshold reaches 3.6% (Models 1-3) and 4.7% (Model 4), with erasure-to-Pauli ratios of 5.0x and 6.5x.
• Per-observable crossing-point analysis at $R_e = 1$ (Models 1-3) independently yields an erasure-to-Pauli ratio of 5.3x.
• The ${10, 3}$ H50 family achieves an erasure-to-Pauli ratio of 5.2x; the ${12, 3}$ H72 family achieves 4.5x.

Interpretation

• Hyperbolic surface codes under erasure noise exhibit erasure thresholds several times higher than their Pauli thresholds, matching the ratio observed for planar surface codes.
• This establishes that the benefits of erasure-aware decoding extend to the hyperbolic code architecture, which trades encoding rate for increased distance.
• Fine-grained "semi-hyperbolic" codes, which interpolate between hyperbolic and Euclidean scaling, also show the erasure advantage.
• The results demonstrate that the erasure threshold advantage is a fundamental property of surface codes, independent of the underlying geometry.

Limitations & Uncertainties

• Simulations are limited to moderate code sizes, with the largest base code having 648 physical qubits.
• Fine-graining data only cover refinement levels up to $f=4$; higher levels were not examined.
• Statistical uncertainty in threshold estimates, especially near the crossing points where logical error rates are 10-30%.

What Comes Next

• Investigate higher refinement levels of the fine-grained codes to see if the erasure pseudothreshold continues to increase.
• Extend the erasure analysis to Floquet-type measurement schedules on these hyperbolic tessellations, connecting the CSS results to the broader hyperbolic Floquet literature.
• Explore other hyperbolic tessellation families and investigate the impact of the specific tessellation geometry on erasure thresholds.