  The color code is a topological quantum error-correcting code supporting a
variety of valuable fault-tolerant logical gates. Its two-dimensional version,
the triangular color code, may soon be realized with currently available
superconducting hardware despite constrained qubit connectivity. To guide this
experimental effort, we study the storage threshold of the triangular color
code against circuit-level depolarizing noise. First, we adapt the Restriction
Decoder to the setting of the triangular color code and to phenomenological
noise. Then, we propose a fault-tolerant implementation of the stabilizer
measurement circuits, which incorporates flag qubits. We show how information
from flag qubits can be used with the Restriction Decoder to maintain the
effective distance of the code. We numerically estimate the threshold of the
triangular color code to be 0.2%, which is competitive with the thresholds of
other topological quantum codes. We also prove that 1-flag stabilizer
measurement circuits are sufficient to preserve the full code distance, which
may be used to find simpler syndrome extraction circuits of the color code.
