From 1D Ising Chains to Rings: Boundary and Frustration Effects in the Quantum Approximate Optimization Algorithm

We study shallow-depth Quantum Approximate Optimization Algorithm (QAOA) on one-dimensional Ising models, contrasting open chains with periodic rings to isolate boundary and frustration effects. On modest-qubit instances with exact classical verification, we report energy error, ground-state success probability, and two-point correlators versus depth p. Unfrustrated cases (ferromagnetic;  anti-FM even-n rings) are near-optimal at p = 1, whereas frustrated odd-n AFM rings require p ⁣≥ ⁣ 2. The results yield concise rules-of-thumb for when more depth is necessary in near-term practice and instruction.