🤖 AI Summary
This work addresses the optimization challenges posed by barren plateaus in parametrized quantum circuit training by introducing the first four-player potential game framework that jointly optimizes trainability, non-stabilizerness, task performance, and hardware overhead. The quantum circuit is modeled as a directed acyclic graph, wherein each player performs append, delete, retype, or reconnect operations to seek an ε-Nash equilibrium. An efficient search algorithm based on block-coordinate ε-Nash residuals evolves circuits within a constrained action space. Experiments on MaxCut K₄ and LiH tasks demonstrate that the generated circuits simultaneously achieve high performance, significantly reduced gate counts, enhanced non-stabilizerness, and effective barren plateau avoidance. On a 2×2 grid topology, the approach repeatedly approaches the theoretical potential upper bound, revealing the underlying Pareto trade-offs among the multiple objectives.
📝 Abstract
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 $δ_\text{Nash}$ certifies that no single player can improve unilaterally. A single weight sweep on MaxCut $K_4$ traces a Pareto frontier from a Clifford endpoint $(M_2/n,\langle H\rangle)=(0,4.00)$ to a non-Clifford endpoint $(0.48,3.30)$. On three four-qubit hardware topologies (heavy-hex, $2\times 2$ grid, Rydberg all-to-all), Nash search achieves the highest mean potential; on the $2\times 2$ grid Nash reaches the theoretical ceiling $Φ_\text{max}=4.10$ 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 ($p\ge 0.22$). 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.