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 $d-\lfloor (d+3)/6\rfloor$, rather than the $d/2$ distance associated with spatially uniform color code scheduling.