A Comparison of Quadratic and Higher-Order Representations for QAOA

In this work we consider a routing problem and compare quadratic and higher-order representations using the Quantum Approximate Optimisation Algorithm (QAOA). The majority of works investigating QAOA use quadratic Hamiltonians to represent the considered problems, which can lead to poor scaling in qubit requirements. We address the gap of direct comparisons between quadratic and higher-order forms through an investigation into two distinct formulations of the same use case. We find that the higher-order form yields better solution quality and scales better in terms of numbers of qubits, but requires more two-qubit gates. We additionally consider a factoring method to reduce the gate depth of the higher-order version, which achieves a significant reduction in the number of two-qubit gates when run on real IBM hardware.