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

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