No Quantum Advantage in Decoded Quantum Interferometry for MaxCut
Ojas Parekh
TL;DR
The paper examines Decoded Quantum Interferometry (DQI) for MaxCut and proves that any instance where DQI offers a nontrivial approximation is actually solvable exactly in polynomial time by classical means. It specializes DQI to MaxCut, showing that the decoding step reduces to a minimum $T$-join problem, solvable via a perfect-matching reduction, and ties the success of DQI to graph girth through an injectivity condition on a map from edge-subgraphs to vertex-parities. A key result is that when the girth $g$ is linear in the input size $m$, MaxCut can be solved efficiently by classical algorithms, implying no quantum advantage in this high-girth regime. The work also presents a streamlined, graph-theoretic presentation of DQI for MaxCut and discusses extensions, limitations, and directions for identifying problems where DQI might offer genuine advantages beyond this regime.
Abstract
Decoded Quantum Interferometry (DQI) is a framework for approximating special kinds of discrete optimization problems that relies on problem structure in a way that sets it apart from other classical or quantum approaches. We show that the instances of MaxCut on which DQI attains a nontrivial asymptotic approximation guarantee are solvable exactly in classical polynomial time. We include a streamlined exposition of DQI tailored for MaxCut that relies on elementary graph theory instead of coding theory to motivate and explain the algorithm.