Transferred QAOA Parameters Remember the Penalty Scale: A $λ$-Resonance Law for Constrained Quantum Optimization

Abstract

Training the variational angles of the Quantum Approximate Optimization Algorithm once on a small instance and reusing them on larger ones, known as parameter transfer, is the standard route past the exact-simulation wall. Existing literature explains its success almost entirely through structural similarity. We identify a new, independent axis that governs transfer whenever constraints are encoded as penalties: the trained angles memorize the penalty weight $λ$ of their training instance. For any scalarized cost function with an integer-valued violation count, we prove that at arbitrary fixed QAOA angles $(β,γ)$ of depth $p$, the probability mass $F(λ)$ on the feasible subspace is a finite real trigonometric polynomial in $λ$ whose angular frequencies lie on an integer lattice generated by the trained $γ$'s. Three consequences follow immediately: transfer feasibility is a resonance peaked where the deployment penalty matches the training penalty; the resonance width scales as $1/(v_{max}\sum_k|γ_k|)$, so low-$|γ|$ angle sets are systematically more transferable; and the curve exhibits revival peaks at spacings $2π/γ_k$. We confirm all three predictions by exact statevector experiments on a 20-qubit multi-user resource-allocation QUBO. The theorem is independent of how the angles were obtained and applies to any integer-penalty QUBO, recasting a widely reported failure mode of penalty-based QAOA as deterministic, predictable phase interference rather than an energetic tuning problem.