Optimization of Quadratic Constraints by Decoded Quantum Interferometry

Abstract

A recent paper by Jordan et al. introduced Decoded Quantum Interferometry (DQI), a novel quantum algorithm that uses the quantum Fourier transform to reduce linear optimization problems -- max-XORSAT and max-LINSAT -- to decoding problems. In this paper, we extend DQI to optimization problems involving quadratic constraints, which we call max-QUADSAT. Leveraging a connection to quadratic Gauss sums, we give an efficient algorithm to prepare the DQI state for max-QUADSAT (DISCLAIMER: A mistake was found in one of the algorithm's steps, which invalidates this result unless a fix is found). To demonstrate that our algorithm achieves a quantum advantage, we introduce the Quadratic Optimal Polynomial Intersection (quadratic-OPI) problem, a restricted variant of OPI for which, to our knowledge, the standard DQI framework offers no algorithmic speedup. We show that quadratic-OPI is an instance of max-QUADSAT and use our algorithm to optimize it. Lastly, we present a new generalized proof of the "semicircle law" for the fraction of satisfied constraints, generalizing it to any DQI state of problems where the distribution of the number of satisfied constraints for a random assignment is sufficiently close to a binomial distribution. This condition holds exactly for the DQI state of max-LINSAT, and approximately holds in the max-QUADSAT case, with the approximation becoming exponentially better as the problem size increases. This establishes performance guarantees for our algorithm.