Pricing the side conditions of a gauged logical measurement

Abstract

Quantum algorithms act on protected information by measuring it, so how much noise that measurement survives sets how much noise a computation can absorb. Gauging implements the measurement by coupling the code to auxiliary qubits, one per edge of a graph built on the operator's support, and turns the circuit into a code in space and time: its fault distance splits into a spatial component, the distance of the code the measurement leaves behind, and a temporal one, the rounds of faults it tolerates undetected. Williamson and Yoder bound both components below by the code distance under four side conditions on the code and an auxiliary graph. The bound consumes two of those conditions and states the component each one bounds, but it prices neither: it proves no necessity statement, and no instance comes with the distance it falls to. The list is our reading: two are hypotheses of one theorem, one a convention, one a lemma's hypothesis. Which entries are doing work, and what the distance becomes when one fails, are open questions. Here we show that of the statement this development assembles only two of the four are consumed, that the round count among them is exactly tight, and that the condition it never consumes, the demand that the first and last rounds be perfect, is what holds the temporal component up. That condition reads like a convention about a circuit's ends, yet deleting the boundary holds the fault distance at one at every round count. The expansion condition is two-sided, one graph showing it is not necessary and another that its failure costs the bound, and on the toric family it needs no per-member decision once the auxiliary graph is complete, at one ancilla per edge. The round-count condition is exactly tight in the measurement-fault model, where one round below its demand the Bacon--Shor transversal measurement admits an undetectable weight-two spacetime fault against a code distance of three. In the fuller protocol, which carries data and ancilla faults too, two rounds already restore it. Both components are computed rather than bounded on named instances, a code distance on the gauged \([[18,4,4]]\) the measurement takes to \([[24,3,4]]\) and a round count on the Bacon--Shor \([[9,1,3]]\), and the counting lemma behind the spatial bound is machine-checked in all three of its steps. The fault-distance figure is machine-checked too, and the chain closes on three standard axioms with no solver; the closed forms are tested by second implementations instead. The question is general: where a guarantee is assembled from a list of side conditions, which entries carry it has an answer, and here all four have one. Three of the conditions are predicates the development evaluates, the fourth is decided by the rank of a check matrix, and its cost turns out to be zero.