Expectation values from the single-layer quantum approximate optimization algorithm on Ising problems

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