Junction Hook Errors in the Color Code
Abstract
Color-dependent syndrome extraction schedules have been proposed to mitigate hook errors in the triangular color code. Recent work by Kishony and Fowler (arXiv:2603.28852) identifies distance-reducing hook errors by the type of anyon for which they provide a shortcut and chooses the CNOT ordering for stabilizers of each color to avoid these errors. We identify an additional class of distance-reducing hooks, which we call junction hook errors, that is not captured by this classification. The two data qubits of a junction hook error terminate anyon strings of two different colors at the point where the three colored strings of a logical operator meet, which we call the junction. As such, the hook error does not correspond to a single type of anyon, yet it still reduces the distance. For distances d=7,9,11, we verify that every non-corner plaquette admits a scheduled hook error that, together with d−2 single data qubit faults, produces an undetectable logical error using d−1 faults. Each of these is a junction hook error. Because of this, every stabilizer, including those in the bulk, still requires additional protection from hook errors (such as flag qubits) to restore the full circuit-level distance with this schedule. A minimum weight logical operator has at most one junction, however, so junction hook errors cannot accumulate on one minimum weight logical operator in the way that single-color malign hook errors can. This helps explain why the schedule still improves the circuit-level distance to d−⌊ (d+3)/6⌋, rather than the d/2 distance associated with spatially uniform color code scheduling.
Related
- Code-Visible Liouvillian Modes and Coherent Logical Errors in Quantum Codes
- Early Fault-Tolerant Magic State Synthesis and Spacetime Efficient Analog Rotations with Color Code on a Superconducting Quantum Processor
- Quantum error correction well below the color code threshold
- Anisometric Dielectric Snowman-Particle Provides Improved Control Over Photonic Hook
- Synchronization Threshold and Stability of Heterogeneous Josephson-Junction Arrays with Nonsymmetric Weighted Coupling
BibTeX
@misc{issa2026junction,
title = {{Junction Hook Errors in the Color Code}},
author = {Salahedeen Issa and Margaret Martonosi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33347},
eprint = {2610.33347},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33347},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Junction Hook Errors in the Color Code
AU - Salahedeen Issa
AU - Margaret Martonosi
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.33347
UR - https://arxiv.org/abs/2610.33347
AB - Color-dependent syndrome extraction schedules have been proposed to mitigate hook errors in the triangular color code. Recent work by Kishony and Fowler (arXiv:2603.28852) identifies distance-reducing hook errors by the type of anyon for which they provide a shortcut and chooses the CNOT ordering for stabilizers of each color to avoid these errors. We identify an additional class of distance-reducing hooks, which we call junction hook errors, that is not captured by this classification. The two data qubits of a junction hook error terminate anyon strings of two different colors at the point where the three colored strings of a logical operator meet, which we call the junction. As such, the hook error does not correspond to a single type of anyon, yet it still reduces the distance. For distances d=7,9,11, we verify that every non-corner plaquette admits a scheduled hook error that, together with d−2 single data qubit faults, produces an undetectable logical error using d−1 faults. Each of these is a junction hook error. Because of this, every stabilizer, including those in the bulk, still requires additional protection from hook errors (such as flag qubits) to restore the full circuit-level distance with this schedule. A minimum weight logical operator has at most one junction, however, so junction hook errors cannot accumulate on one minimum weight logical operator in the way that single-color malign hook errors can. This helps explain why the schedule still improves the circuit-level distance to d−⌊ (d+3)/6⌋, rather than the d/2 distance associated with spatially uniform color code scheduling.
ER -