Plateau-Constrained Selection: Exploiting Degeneracy for Lower-Depth Quantum Compilation
Minimum-cost orders of commuting phase terms can produce substantially different routed circuits. On the same 36-term instances, three orders with identical support cost 74 yield mean routed depths of 228.6, 233.8, and 256.7. We exploit this degeneracy under fixed placement and maintained-parity lowering: Stage 1 attains the support optimum, and Stage 2 selects minimum routed depth among 24 equal-cost orders. For distinct pair supports, we characterize orders attaining the support lower bound through Hamiltonian paths of the support line graph and count optima exactly through 20 terms. On synthetic 16-qubit assignment-Ising instances, selection reduces depth by 12.83% under a different SABRE routing seed, with lower depth in all 20 instances. The selected orders lie a median 1.57 pool standard deviations below the pool mean, consistent with ordinary best-of-24 selection; the useful feature is that candidate rankings persist across routing seeds. Depth reductions extend to 48 terms and a random-MaxCut generator, whereas evaluation with BasicSwap reverses the gain. A 40-instance IBM Heron study measures a 0.59% error reduction on the executed stabilizer-probe panel, but the primary confidence interval across instances includes zero. Plateau selection therefore improves routed depth in the tested SABRE pipeline while preserving the logical support optimum.