A nearly linear-time Decoded Quantum Interferometry algorithm for the Optimal Polynomial Intersection problem

Recently, Jordan et al. (Nature, 2025) introduced a novel quantum-algorithmic technique called Decoded Quantum Interferometry (DQI) for solving specific combinatorial optimization problems associated with classical codes. They presented a constraint-satisfaction problem called Optimal Polynomial Intersection (OPI) and showed that, for this problem, a DQI algorithm running in polynomial time can satisfy a larger fraction of constraints than any known polynomial-time classical algorithm. In this work, we propose several improvements to the DQI algorithm, including sidestepping the quadratic-time Dicke state preparation. Given random access to the input, we show how these improvements result in a nearly linear-time DQI algorithm for the OPI problem. Concurrently and independently with this work, Khattar et al. (arXiv:2510:10967) also construct a nearly linear-time DQI algorithm for OPI using slightly different techniques.