The cost of each side condition in a gauged logical measurement

Abstract

A logical measurement reads out protected information; the noise it survives sets how much noise a quantum computation can absorb. Gauging builds such a measurement from an auxiliary graph, one auxiliary qubit per edge, turning a global operator into local parity checks. Its tolerance splits in two: the distance of the deformed code, the fewest qubits a fault must touch to go unseen, and the rounds a fault can hide in. Williamson and Yoder proved both components stay at or above the original code's distance, under four side conditions, extra hypotheses assumed as a package. None of those conditions is priced. Here we show the entries are not worth the same: the demand that the first and last rounds be perfect, which reads like bookkeeping, is what carries the temporal part, the rounds a fault can hide in. Dropping it holds the fault distance at 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 give one and two. The round count is tight in the model where comparing rounds is the whole rule, and the full protocol does not need it: the distance returns a round early. Both components are computed exactly, on a gauged bicycle code and a Bacon--Shor measurement, inside a proof assistant. Threshold estimates consume numbers of this kind; side conditions can be priced.