Evidence that PUBO outperforms QUBO when solving continuous optimization problems with the QAOA

TL;DR

This paper demonstrates that PUBO formulations generally outperform QUBO in solving continuous optimization problems with QAOA, especially when native multi-qubit gates are available, leading to better results with fewer qubits.

Contribution

The study provides an extensive evaluation showing PUBO's advantages over QUBO for continuous problems in quantum optimization, highlighting the impact of multi-qubit gates.

Findings

PUBO yields better results than QUBO in continuous optimization.

PUBO requires fewer qubits than QUBO.

Native multi-qubit gates enhance PUBO's performance for higher order problems.

Abstract

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.