Optimization of Quadratic Constraints by Decoded Quantum Interferometry
Daniel Cohen Hillel
TL;DR
This work generalizes Decoded Quantum Interferometry (DQI) to optimization problems with quadratic constraints, introducing max-QUADSAT and linking the DQI state to quadratic Gauss sums. It provides an efficient state-preparation algorithm for diagonal quadratic constraints (no linear terms) and demonstrates quantum advantage via Quadratic Optimal Polynomial Intersection (quadratic-OPI), a max-QUADSAT instance closely related to OPI. A generalized semicircle-law characterization for the fraction of satisfied constraints is proved, ensuring performance guarantees by showing the distribution of satisfied constraints remains close to a binomial model and that the mean is preserved. The results broaden the class of problems amenable to DQI, with potential future extensions to non-diagonal constraints and mixed linear-quadratic formulations.
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. 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.