A provable quantum advantage for approximate optimization via decoded quantum interferometry

Abstract: Decoded quantum interferometry (DQI) is a novel paradigm for tackling approximate optimization problems on quantum computers. This framework comes with strong performance guarantees and exploits a well-established duality between optimization and coding theory. A central question, however, is whether DQI can actually provably outperform all polynomial-time classical algorithms. In this work, we establish such an advantage in an oracle setting: we consider an optimization task called folded optimal polynomial intersection (folded OPI), where the acceptance sets are chosen randomly and accessed through membership oracles. We establish a strict gap between the approximation ratio achievable by any polynomial-time classical algorithm and the approximation ratio achieved by the DQI algorithm. Our proof builds on Jordan et al.'s DQI framework for approximate optimization and extends the classical lower-bound method underlying Yamakawa and Zhandry's exact-search oracle separation to approximation. Building on recent developments by Sun and Wootters, Horinaga and Yamakawa, and Jo, we further show that a modified version of the DQI algorithm achieves a strictly larger gap on the folded OPI problem, yielding an even stronger quantum separation. As a concrete example, for code rate $0.3$, DQI and the modified algorithm achieve expected scores of approximately $0.85$ and $0.95$, respectively. In contrast, exceeding the classical threshold of $0.65$ by any fixed amount with constant probability on sampled instances requires super-polynomially many classical membership queries.