This paper proves a clear quantum advantage for a specific approximate optimization task in an oracle setting. The authors study a framework called decoded quantum interferometry (DQI) and show that, for a family of balanced instances built from folded Reed–Solomon codes, a quantum algorithm can achieve a strictly better approximation score than any polynomial-time classical algorithm that uses only membership queries to the instances.

The optimization task is a version of “folded optimal polynomial intersection” (folded OPI). Each instance is split into many blocks. For each block there is an acceptance set (the part of the block alphabet that counts as a success) and an oracle that answers whether a given symbol is in that set. The goal is to find a candidate polynomial whose folded evaluations are accepted on as many blocks as possible; the score is the fraction of accepting blocks. In the balanced setting studied here, each block accepts exactly half of its alphabet. The family is indexed by a fixed code rate R (a parameter of the Reed–Solomon code) and the number of blocks.

What the researchers did is use the DQI framework to turn a coherent decoder for the dual code into a quantum optimization procedure. Concretely, the DQI algorithm uses one coherent query to each block’s membership oracle and a standard Reed–Solomon unique decoder (implemented via the Berlekamp–Massey algorithm) to produce a candidate with an expected score that is provably above the classical threshold. The classical threshold is the score achieved by an information-set strategy of Prange, which classical algorithms can reach with linearly many membership queries. The paper proves that any classical algorithm that beats that threshold by a fixed amount, with constant probability over sampled instances, would need super-polynomially many membership queries.