September 5, 2026
A frequency‑collision model that ignores gate direction reports 27 violations on a working IBM processor. The real number is 4.
The finding
The standard collision taxonomy is enumerated over coupled pairs, but three of its seven conditions need a control driving two or more targets. Reading the coupling map as undirected admits 195 spectator configurations at 127 qubits where 51 physically exist, and turns two shipping IBM Eagle devices into chips that look broken. A fourth condition, the cross-resonance straddling window, fires on 39.6% of one device's real gates.
Scaling fixed-frequency transmons is limited by frequency collisions, and the standard taxonomy for them is Hertzberg et al.'s seven conditions (arXiv:2009.00781). We implemented it, ran it against the measured frequencies and real gate sets of two working IBM Eagle r3 processors, and got two things wrong on the way. Both are easy to get wrong independently, which is why they are worth writing down.
Everything below is reproducible from public calibration data. No account, no partnership, nothing of ours required.
Direction is load-bearing
Conditions 5 through 7 involve a spectator: a third qubit coupled to the control and disturbed by a gate it is not part of. A spectator configuration needs one control driving two or more targets. Enumerate from undirected adjacency instead and you admit configurations the hardware never runs — most obviously a qubit that controls the middle one being counted as a spectator of its gate.
| at 127 qubits | |
|---|---|
| Triples from undirected adjacency | 195 |
| Triples that physically exist (ibm_brisbane) | 51 |
| Triples that physically exist (ibm_sherbrooke) | 48 |
The real structure is far sparser than adjacency suggests: only 99 of 127 qubits are ever a control on brisbane, and only 39 of those drive more than one gate.
The consequence is not a rounding error. Our undirected version reported 27 and 26 collisions on these two devices, which reads as evidence that the industry's standard collision criterion is meaningless. The directed version reports 4 and 1, with zero nearest-neighbour violations on either. The rule set was fine. The enumeration was not, and the first conclusion we drew from it was dramatic, confident and wrong.
A useful side effect: Hertzberg's Table 1 carries a second count — 8, 25, 51 for 23, 65, 127 qubits — that the paper never defines. It is the spectator triple count. We computed 51 independently from brisbane's real control roles and it matches exactly.
The straddling condition is not a collision
Condition 4 asks the target to sit inside the control's straddling window,
f_c + α_c < f_t < f_c. Against the real directed gate sets:
| Device | Condition 4 fires | Of which target above control |
|---|---|---|
| ibm_brisbane | 57 / 144 (39.6%) | 51 |
| ibm_sherbrooke | 27 / 144 (18.8%) | 26 |
Measured control–target detunings run from −205 to +425 MHz on brisbane and −400 to +365 MHz on sherbrooke. IBM runs ECR in both orientations relative to frequency, routinely, on processors that ship and work.
None of this contradicts the literature — cross-resonance outside the straddling regime is well established, and arXiv:2605.07868 analyses collision conditions specifically for it. The point is narrower and practical: condition 4 sits in the same numbered list as the other six and is easy to implement as an equal, fatal condition. Do that and your checker rejects 40% of the gates on a device that works.
Because the direction labels carry the whole argument, we checked them three
ways. Qiskit's FakeSherbrooke.target['ecr'] agrees with our parse on 144 of 144
directed pairs, zero reversed. Condition 2 is directional and fires 0 times as
labelled against 2 reversed. Condition 4 fires 57 times as labelled against 93
reversed — the labelled direction is far more straddling-compliant, which is what
a designer would choose.
What we could not resolve
Running the reference model's own setup we report 4.3% collision-free yield at 127 qubits and σ_f = 14 MHz, where the published value is 8%. Pessimistic, and growing with size: yield ratio 0.85 at d=3, 0.62 at d=5, 0.54 at d=7.
Three explanations are excluded. The spectator over-count above — fixed, triples fell 195 → 71, and the yield ratio did not move at all, which falsifies our own prior hypothesis that it was the main cause. A condition-2 implementation error — verified correct: 72 of 143 gates sit exactly 50 MHz from the condition-2 resonance, predicting 2.6 violations against 2.64 observed. And a mis-set condition-2 bound — rescaling it alone overshoots, landing at 2.29 against 2.7.
What remains is something in how the reference Monte Carlo is specified. We have not closed it.
What we are claiming, and what we are not
We are not claiming a defect in an IBM device. Both findings are defects in a model of those devices, and both were ours before they were anyone else's. The devices come out of this looking better than the model did.
We have no measured wafer yield. Collisions can be counted on chips that shipped; the chips that did not ship are not observable, and nothing here is a yield measurement.
Two devices, one architecture, one snapshot: Eagle r3, heavy-hex, ECR, 2025-04-30. The collision windows are Hertzberg's estimates for bus-coupled transmons near α = −330 MHz, not constants.
If you are building a checker against this taxonomy: enumerate spectators from the directed gate set, and do not let condition 4 fail a design on its own.