A four-player potential game for barren-plateau-aware quantum ansatz design

We cast the design of parameterized quantum circuits as a four-player potential game whose state is a circuit directed acyclic graph (DAG) and whose players encode trainability, non-stabilizerness, task performance, and hardware cost. Per-player restricted action sets factorize the move space into append, remove, retype, and rewire operations; a block-coordinate varepsilon-Nash residual δ_Nash certifies that no single player can improve unilaterally. A single weight sweep on MaxCut K₄ traces a Pareto frontier from a Clifford endpoint \(M₂/n,langle Hrangle\)=(0,4.00) to a non-Clifford endpoint (0.48,3.30). On three four-qubit hardware topologies heavy-hex, $2times 2$ grid, Rydberg all-to-all, Nash search achieves the highest mean potential; on the 2times 2 grid Nash reaches the theoretical ceiling Φ_max=4.10 on two of five seeds while the simulated-annealing baseline does so on one; paired Wilcoxon tests over five seeds cannot reject the null on any single topology $pge 0.22$. On LiH/STO-3G, seeding Nash from a 58-gate Givens-doubles ansatz produces a 48-operation, depth-25 circuit retaining 97.7\% of the correlation energy while simultaneously reducing gate count, increasing non-stabilizerness, and controlling trainability. The framework is complementary to energy-only searches such as ADAPT-VQE and k-UpCCGSD, which reach chemical accuracy with fewer operations but do not optimize the other three axes.