The cost of each side condition in a gauged logical measurement

Abstract

A logical measurement reads out protected information, and the noise it survives sets how much a quantum computation can absorb. Gauging builds one from a graph on the measured qubits, one extra qubit on each edge, turning a global operator into local parity checks. Its tolerance splits in two: the distance of the code left behind, and the rounds a fault can hide in. Williamson and Yoder proved both components stay at or above the code's distance. The guarantee rests on four side conditions taken as a package, none of them priced. Here we show they are not worth the same: the demand that the first and last rounds be perfect, which reads like bookkeeping, carries the temporal part. Dropping it flattens the fault distance, the least weight of a fault that passes unseen and flips the readout, to one for every code and round count, in a model whose detectors compare adjacent rounds; the full protocol collapses on the instance measured. The expansion condition, that no small set of qubits be sealed off, does not decide the outcome: two paths on the same qubits, both outside the covered regime, give distances one and two. The round count is tight where comparing rounds is the whole rule; the full protocol reaches the distance a round early. Both components are computed exactly on a gauged bivariate bicycle code and a Bacon--Shor measurement, inside a proof assistant. Threshold estimates consume such numbers; side conditions can be priced.