Quantum Decoding Algorithms: Quantum Speedups in Optimization

TL;DR

This paper reviews Decoded Quantum Interferometry (DQI), a quantum algorithm promising superpolynomial speedups for certain optimization problems like max-LINSAT and OPI, using coding theory and interferometry.

Contribution

It provides a comprehensive, self-contained overview of DQI, detailing its background, algorithmic steps, and potential for quantum speedups in practical optimization.

Findings

DQI shows strong evidence of superpolynomial speedup for OPI.

The paper offers a detailed explanation of the DQI algorithm and its theoretical basis.

It connects coding theory and interferometry to quantum optimization techniques.

Abstract

Attaining a quantum speedup in solving practically useful optimization problems has been one of the holy grails in the field of quantum computing. While prior approaches have demonstrated speedups for certain structured problem classes, establishing a clear and scalable advantage on broadly useful practical optimization problems remains challenging. Recently, a new approach to solving the max-LINSAT class of optimization problems has emerged, called Decoded Quantum Interferometry (DQI). In DQI, a combination of techniques rooted in (classical) coding theory and interferometry are used to obtain the solution of max-LINSAT. In the special problem instance of the optimal polynomial intersection (OPI) problem, strong evidence exists to show that an superpolynomial speedup exists over the best classical methods in obtaining an approximate solution. In this review, we give a self-contained description of DQI and the necessary background to understand the algorithm. Specifically, we give the essentials of Galois fields, optimization problems such as max-LINSAT and OPI, and coding theory, followed by a step-by-step walkthrough of the quantum algorithm and its operating principle.