Folding a Ring onto Heavy-Hex: Formula-Built Schedules with Provable Gate Counts, and What Their Signal Shows on Twins and Devices

A ring of n CZ gates per layer, the entangling pattern of many variational classifiers, doesnot fit IBM’s heavy-hex devices unless n is a cycle length of the lattice, and heavy-hex hascycles of 12,20,24,28,... qubits only. We compare three schedules that a compiler can buildby formula, with no layout search and without reading the day’s calibration: the zipper (thering folded onto a line; 5n/2 − 4 two-qubit gates per layer for even n, (5n − 7)/2 for odd n),holes (the ring on a longer cycle, its spare qubits moved as |0⟩; n + 2(m − n)), and pendants(the ring on a shorter cycle, its extra nodes hung off it; n + (n − m), on exactly n qubits). Weprove that the pendant schedule’s count is the least possible for layered schedules on the cycleand its leaves in a stated routing model (exhaustive search finds the same for interleaved onesin small cases), that the hole schedule’s count is least within its class but not in general, andrecall that no cheaper schedule than the zipper was found by exhaustive searches at n = 4–12and 14. A rule that takes the fewest extra gates picks pendants at n = 13–17 and holes atn = 8–11 and 18–19 on IBM’s Heron devices; the rule is in PSF-Zero’s AI front end a26. Onnoise models, in declared matches against the zipper at 16 layers over four random classifiers,the rule won at every n = 13–18 on two twins. Its lead was 1.5–8.2× the zipper’s kept signal onthe twin of a 13-day-old snapshot and only 1.07–1.28× (about even at n = 18) on the twin of areal day’s calibration, the kind that had predicted the zipper on hardware: the old snapshot waspessimistic for the zipper. On two devices (ibmkingston, ibmfez) the pendant schedule keptthe signal its day’s twin predicted (0.90±0.12 against 0.77; 0.50±0.10 against 0.59), in two jobsof 7s each; these are coarse tests and no device comparison with the zipper was made. AgainstQiskit’s search (level 2 on the same twin) the rule won on every classifier at every n simulated(n = 13–16, 18): Qiskit’s circuits kept −0.05–0.64 of the signal against 0.61–0.82. The bare ringkept 0.78–0.86 at n = 20 and 0.81–0.83 at n = 24. On one 20-qubit cycle, two pendants keptnearly what the bare ring kept (0.75–0.84 against 0.78–0.86) and clearly more than two holes(0.61–0.68); with the classifiers and qubits fixed and only the schedule changed, the bare ringkept 1.44–1.81 times the zipper’s signal and two holes 1.13–1.20 times. Error-product estimatesof the kept signal did not rank the schedules. We give the benchmark number and the deviceuser’s number side by side throughout