Extrapolation method to optimize linear-ramp QAOA parameters: Evaluation of QAOA runtime scaling

The Quantum Approximate Optimization Algorithm (QAOA) has been suggested as a promising candidate for the solution of combinatorial optimization problems. Yet, whether - or under what conditions - it may offer an advantage compared to classical algorithms remains to be proven. Using the standard variational form of QAOA requires a high number of circuit parameters that have to be optimized at a sufficiently large depth, which constitutes a bottleneck for achieving a potential scaling advantage. The linear-ramp QAOA (LR-QAOA) has been proposed to address this issue, as it relies on only two parameters which have to be optimized. Based on this, we develop a method to estimate suitable values for those parameters through extrapolation, starting from smaller problem sizes (number of qubits) towards larger problem sizes. We apply this method to several use cases such as portfolio optimization, feature selection, clustering and weighted maxcut. From results obtained on a noiseless quantum emulator, we evaluate the quantum runtime scaling for finding the optimal solution and compare it with that of classical methods. In the case of portfolio optimization, we demonstrate superior scaling compared to the classical runtime for the problem sizes of up to $28$ qubits that we consider in this work.