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

AbstractA ring of n CZ gates per layer, the entangling pattern of many variational classi ers, doesnot t 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 build byformula, with no layout search and without reading the day’s calibration: the zipper (the ringfolded 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 ringon a shorter cycle, its extra nodes hung o it; n+(n−m), on exactly n qubits). We prove thatthe pendant schedule’s count is the least possible for layered schedules in a stated routing model(exhaustive search nds the same for interleaved ones in small cases), that the hole schedule’scount is least within its class but not in general, and recall that no cheaper schedule than thezipper was found by exhaustive searches at n = 4 12 and 14. A rule that takes the fewest extragates picks pendants at n = 13 17 and holes at n = 8 11 and 1819 on IBM’s Heron devices;the rule is in PSF-Zero’s AI front end a26. On noise models, in declared matches against thezipper at 16 layers over four random classi ers, the rule won at every n = 13 18 on two twins.Its lead was 1.58.2× the zipper’s kept signal on the twin of a 13-day-old snapshot and only1.071.28× (about even at n = 18) on the twin of a real day’s calibration, the kind that hadpredicted the zipper on hardware: the old snapshot was pessimistic for the zipper. On twodevices (ibm_kingston, ibm_fez) the pendant schedule kept the signal its day’s twin predicted(0.90±0.12 against 0.77; 0.50±0.10 against 0.59), in two jobs of 7s each; these are coarse testsand no device comparison with the zipper was made. Error-product estimates of the kept signaldid not rank the schedules. We give the benchmark number and the device user’s number sideby side throughout.