OPI x Soft Decoders

TL;DR

This paper unifies and improves quantum algorithms for the Optimal Polynomial Intersection problem by integrating structured code decoders, simplifying analysis, and extending results under Bernoulli noise models.

Contribution

It reconciles previous approaches to quantum code decoding for OPI, introduces a code-based formulation of recent reductions, and enhances algorithms with stronger soft decoders.

Findings

Recovered Jordan et al.'s results under Bernoulli noise models.

Integrated stronger soft decoders into the OPI framework.

Achieved improved quantum algorithms for OPI.

Abstract

In recent years, a particularly interesting line of research has focused on designing quantum algorithms for code and lattice problems inspired by Regev's reduction. The core idea is to use a decoder for a given code to find short codewords in its dual. For example, Jordan et al. demonstrated how structured codes can be used in this framework to exhibit some quantum advantage. In particular, they showed how the classical decodability of Reed-Solomon codes can be leveraged to solve the Optimal Polynomial Intersection (OPI) problem quantumly. This approach was further improved by Chailloux and Tillich using stronger soft decoders, though their analysis was restricted to a specific setting of OPI. In this work, we reconcile these two approaches. We build on a recent formulation of the reduction by Chailloux and Hermouet in the lattice-based setting, which we rewrite in the language of codes. With this reduction, we show that the results of Jordan et al. can be recovered under Bernoulli noise models, simplifying the analysis. This characterization then allows us to integrate the stronger soft decoders of Chailloux and Tillich into the OPI framework, yielding improved algorithms.