Phi-Based Quantum-Classical Routing: Empirical Evidence for a Crossover Regime on IBM Quantum Hardware

We investigate whether a training-free thermodynamic stability metric Φ = I × ρ - α × S, computed from IBM Quantum calibration data before circuit execution, can serve as a routing signal for allocating computation between quantum hardware and classical simulation in the tested setting. Across 23 tests on real IBM Quantum hardware, we validate a Φ-based routing framework and show in direct crossover tests that LOW-Φ quantum execution can be worse than exact classical simulation for the tested classically tractable circuits: in the primary crossover test, classical simulation achieved 0.00% error while LOW-Φ quantum execution produced 8.23% error on the same circuit. Under a strict methodology with verified physical qubit binding and decisive ground truth, HIGH-Φ versus LOW-Φ error ratios ranged from 2.90x (QPE) to 30.47x (Bernstein-Vazirani) across 7 tested algorithms. The strict seven-algorithm tests constitute the primary evidence layer, while the broader 23-test suite provides supporting boundary conditions on size, depth, and reproducibility. Circuit size and depth impose additional routing boundaries: in the tested conditions, LOW-Φ execution at 15 qubits produced 94.85% error and execution at circuit depth 103 produced 67.29% error, both transitions into regimes where classical fallback was more reliable. These results support a routing decision framework in which minimum Φ across circuit qubits determines whether to execute on quantum hardware or fall back to classical simulation, with hybrid strategies retained as a natural extension. The methods described in this paper are the subject of U.S. Provisional Patent Application No. 63/956,752 (filed January 9, 2026). No license to implement or commercialize the described methods is granted by this publication. All rights reserved.