A quantum code's threshold is not a property of the code. It is a property of the code, the syndrome measurement circuit that extracts its checks, and the classical decoder that interprets them, evaluated jointly under a noise model where every operation can fail. Fixing the acceptance criteria before choosing the construction is what makes the comparison honest: encoding rate against code distance, syndrome circuit depth, pseudo-threshold measured against the break-even point of not encoding at all, Tanner-graph thickness as a hardware buildability constraint, and the presence of load-store and individual-qubit readout. Judged this way, a low-density parity-check code with non-local weight-six checks holds twelve logical qubits in 288 physical ones at a threshold matching the surface code, where surface-code patches would need roughly ten times the hardware - a result invisible to any comparison of codes in the abstract.
Sergey Bravyi, Andrew W. Cross, Jay M. Gambetta, Dmitri Maslov and Theodore J. Yoder (IBM Quantum, T.J. Watson Research Center) with Patrick Rall (IBM Quantum, MIT-IBM Watson AI Lab), arXiv 2308.0791…