Experimental supplement: three-qubit, three-gate repetition-code noise tomography

Experimental supplement to arXiv:2611.08310, a companion manuscript on gate-optimal recovery designs (2026). This supplement specifies a concrete experiment for measuring the eleven-parameter small-angle coherent noise of a three-qubit register with three single-qubit Pauli gates, using a distance-three repetition code in both orientations. It gives the branch structure, the syndrome-to-Pauli look-up table for the three-gate design, the fixed sixteen-setting measurement table, shot and wall-clock budgets for three platform classes, a calibration and systematics checklist, an analysis pipeline, and a symptom-to-cause table. Two sections price the alternatives: a fixed recovery that does not resolve the syndrome saves a factor three in circuits and loses a factor five in precision floor, and a coherent unitary recovery keeps the circuit saving but replaces each single-qubit Pauli by a multi-controlled gate that every branch must compile. At three qubits no design in the scanned budget loses rank, so identifiability does not separate the gate counts; what depends on the working point is which three gates to use, not how many. The organising point is that the recovery gate count is a resource: three gates attain the lowest precision floor, the standard six-gate decoder is 22 times worse at a small-angle operating point and 2.3 times worse at the well-conditioned point of arXiv:2610.19779, and the ordering is stable over forty sampled working points while the size of the factor is not. The ordering survives when the look-up is compiled rather than measured, and the margin then widens from 5.6 to 7.7 in wall clock, because a compiled table charges every entry in every branch. A section added in version 1.5 asks whether that answer depends on the platform. A second device archetype -- a tunable coupler or iSWAP-style gate, whose always-on term is the exchange J(XX+YY)/2 -- carries one more parameter per bond, thirteen in all, and the two clusters are not nested: XX and YY commute, and ZZ is orthogonal to their plane. Enumerating all 64 design pairs at n=3 and all 3969 at n=5 gives the same gate budget for both archetypes, zero gates to reach full rank at n=3 and one at n=5; that count is a lower bound on the budget, not a recommended design, since the precision optimum is the three gates of the main text, where zero gates is 18 times worse at the median and 379 times worse at the certified point. At matched per-layer coherent infidelity the thirteen-parameter cluster is not worse (101.0 against 88.9 mrad on the local angles at 10^5 shots), and at the three-gate optimum the comparison widens (41.7 against 32.2 mrad). Choosing the wrong cluster is a bias, not a variance: the correct family leaves a residual of exactly zero, the wrong one leaves 0.4-0.6% in the fingerprint and moves the nine local angles by a quarter to a half, while its Cramer-Rao bound looks better. The decoded logical error cannot arbitrate at that bias level -- 200 random perturbations of the same total variation spread it over a factor 43 -- so the model family must be chosen by the residual. The same section closes with a composition law: a per-gate composition of Pauli probabilities is elementwise in the Walsh eigenvalues and therefore under-counts the coherent part of the depth-L noise by exactly L (measured slopes 1.98 for the generator against 0.99 for the per-gate rule). A closing section fixes the size of the gain and asks how far it extends. An exact prior removes 19.3% of the logical error at d=3, which is 4.3e-5 errors per code cycle; a three-gate fit captures 83, 96.5 and 99.5% of that share on the three platform classes. That share, not the accuracy of the estimate, is where every prior-based protocol stops, because 94% of the residual is weight-two aliasing. It is not a limit set by using Pauli probabilities: the same prior decodes both the coherent channel and its twirled version, so the drop from 19.3% to 11.6% under twirling prices the device rather than the estimate, and composing per-gate twirls changes at most two of the 256 decoder decisions. A generator rather than a channel is what additionally allows the coherent part to be inverted and the prior to be extrapolated to unmeasured working points. Carrying the same three-qubit fit to d=5 or d=7 obeys a closed form, G_d = G_3^((d+1)/4), so no extra precision is needed at larger distance: a 6.4 mrad rms on the eleven parameters moves the d=7 gain by 8.5%. What does not transport is the truncated part of the model---the 2(d-1)^2 three-body positions a d x d patch adds, none of which a chain fit covers---which costs 5% of the gain at d=5 and 10% at d=7; site-to-site spread costs another 3%. The law itself is a property of the decoder and applies to any prior, so this is a bound on what a three-qubit run can claim, not an advantage over Pauli-statistics protocols. Measuring at the target size instead of transporting avoids the truncation count but pays conditioning: a factor 68 from three to seven qubits. All numbers are model-level: no device data are reported, and the floors assume a calibration residual of 1e-3. Contents (v1.5): 1 What the experiment measures · 2 The three branches · 3 Syndrome decoding · 4 Settings and shot arithmetic · 5 Budget on three platforms · 6 Why the syndromes must be resolved · 7 What does and does not depend on the gate count · 8 Does the platform change the answer? · 9 How large the gain is, and how far it extends · 10 Calibration and systematics · 11 Analysis pipeline · 12 Diagnostics · 13 Checklist.