  Quantum computing provides powerful algorithmic tools that have been shown to
outperform established classical solvers in specific optimization tasks. A core
step in solving optimization problems with known quantum algorithms such as the
Quantum Approximate Optimization Algorithm (QAOA) is the problem formulation.
While quantum optimization has historically centered around Quadratic
Unconstrained Optimization (QUBO) problems, recent studies show, that many
combinatorial problems such as the TSP can be solved more efficiently in their
native Polynomial Unconstrained Optimization (PUBO) forms. As many optimization
problems in practice also contain continuous variables, our contribution
investigates the performance of the QAOA in solving continuous optimization
problems when using PUBO and QUBO formulations. Our extensive evaluation on
suitable benchmark functions, shows that PUBO formulations generally yield
better results, while requiring less qubits. As the multi-qubit interactions
needed for the PUBO variant have to be decomposed using the hardware gates
available, i.e., currently single- and two-qubit gates, the circuit depth of
the PUBO approach outscales its QUBO alternative roughly linearly in the order
of the objective function. However, incorporating the planned addition of
native multi-qubit gates such as the global Molmer-Sorenson gate, our
experiments indicate that PUBO outperforms QUBO for higher order continuous
optimization problems in general.
