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We report on the energy-expectation-value landscapes produced by thesingle-layer ($p=1$) Quantum Approximate Optimization Algorithm (QAOA) whenbeing used to solve Ising problems. The landscapes are obtained using ananalytical formula that we derive. The formula allows us to predict thelandscape for any given Ising problem instance and consequently predict theoptimal QAOA parameters for heuristically solving that instance using thesingle-layer QAOA. We have validated our analytical formula by showing that itaccurately reproduces the landscapes published in recent experimental reports.We then applied our methods to address the question: how well is thesingle-layer QAOA able to solve large benchmark problem instances? We used ouranalytical formula to calculate the optimal energy-expectation values forbenchmark MAX-CUT problems containing up to $7\,000$ vertices and $41\,459$edges. We also calculated the optimal energy expectations for general Isingproblems with up to $100\,000$ vertices and $150\,000$ edges. Our resultsprovide an estimate for how well the single-layer QAOA may work when run on aquantum computer with thousands of qubits. In addition to providing performanceestimates when optimal angles are used, we are able to use our analyticalresults to investigate the difficulties one may encounter when running the QAOAin practice for different classes of Ising instances. We find that depending onthe parameters of the Ising Hamiltonian, the expectation-value landscapes canbe rather complex, with sharp features that necessitate highly accuraterotation gates in order for the QAOA to be run optimally on quantum hardware.We also present analytical results that explain some of the qualitativelandscape features that are observed numerically.

Expectation Values from the Single-Layer Quantum Approximate Optimization Algorithm on Ising Problems