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.

The paper examines whether color-dependent CNOT schedules for triangular color-code syndrome extraction eliminate all distance-reducing hook errors in the bulk. It identifies a missing class, “junction hook errors,” in the schedule proposed in arXiv:2603.28852. These weight-two errors connect endpoints of two differently colored strings where three strings of a minimum-weight logical operator meet, rather than shortening a single-color string. For $d=7,9,11$, enumeration finds that every non-corner stabilizer admits a scheduled junction hook contained in a weight-$d$ logical operator. Combining that hook with $d-2$ single-data-qubit faults produces an undetectable logical error using $d-1$ faults. These constructions are checked in the supplied syndrome-extraction circuits using Stim. On each weight-six stabilizer, exactly one of its two weight-two hooks belongs to a minimum-weight logical operator; weight-four boundary stabilizers also admit distance-reducing hooks. Exact integer-program calculations with noise restricted to bulk CNOTs and individual data qubits give circuit-level distances $6$ at $d=7$ and $8$ at $d=9$. Consequently, protecting only boundary stabilizers cannot restore full distance with this schedule; bulk hooks also require protection. The previously reported overall distance $d-\lfloor(d+3)/6\rfloor$ is unchanged. A minimum-weight logical operator has at most one junction, limiting accumulation of junction shortcuts; additional distance reduction arises from boundary-hook combinations. Exhaustive hook checks and exact restricted-noise distance calculations cover only the finite distances stated, and the paper does not provide a replacement extraction schedule.