Sum-of-Squares inspired Quantum Metaheuristic for Polynomial Optimization with the Hadamard Test and Approximate Amplitude Constraints

Abstract: Quantum computation shows promise for addressing numerous classically intractable problems, such as optimization tasks. Many optimization problems are NP-hard, meaning that they scale exponentially with problem size and thus cannot be addressed at scale by traditional computing paradigms. The recently proposed quantum algorithm arXiv:2206.14999 addresses this challenge for some NP-hard problems, and is based on classical semidefinite programming (SDP). In this manuscript, we generalize the SDP-inspired quantum algorithm to sum-of-squares programming, which targets a broader problem set. Our proposed algorithm addresses degree-$k$ polynomial optimization problems with $N \leq 2^n$ variables (which are representative of many NP-hard problems) using $O(nk)$ qubits, $O(k)$ quantum measurements, and $O(\textrm{poly}(n))$ classical calculations. We apply the proposed algorithm to the prototypical Max-$k$SAT problem and compare its performance against classical sum-of-squares, state-of-the-art heuristic solvers, and random guessing. Simulations show that the performance of our algorithm surpasses that of classical sum-of-squares after rounding. Our results further demonstrate that our algorithm is suitable for large problems and approximates the best known classical heuristics, while also providing a more generalizable approach compared to problem-specific heuristics.