Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantum advantage from soft decoders

André Chailloux, Jean-Pierre Tillich

TL;DR

A novel generic reduction from a syndrome decoding problem to a coset sampling problem, providing a powerful and simple to use theorem, which generalizes previous work and is of independent interest.

Abstract

In the last years, Regev's reduction has been used as a quantum algorithmic tool for providing a quantum advantage for variants of the decoding problem. Following this line of work, the authors of [JSW+24] have recently come up with a quantum algorithm called Decoded Quantum Interferometry that is able to solve in polynomial time several optimization problems. They study in particular the Optimal Polynomial Interpolation (OPI) problem, which can be seen as a decoding problem on Reed-Solomon codes. In this work, we provide strong improvements for some instantiations of the OPI problem. The most notable improvements are for the $ISIS_{\infty}$ problem (originating from lattice-based cryptography) on Reed-Solomon codes but we also study different constraints for OPI. Our results provide natural and convincing decoding problems for which we believe to have a quantum advantage. Our proof techniques involve the use of a soft decoder for Reed-Solomon codes, namely the decoding algorithm from Koetter and Vardy [KV03]. In order to be able to use this decoder in the setting of Regev's reduction, we provide a novel generic reduction from a syndrome decoding problem to a coset sampling problem, providing a powerful and simple to use theorem, which generalizes previous work and is of independent interest. We also provide an extensive study of OPI using the Koetter and Vardy algorithm.

Quantum advantage from soft decoders

TL;DR

A novel generic reduction from a syndrome decoding problem to a coset sampling problem, providing a powerful and simple to use theorem, which generalizes previous work and is of independent interest.

Abstract

In the last years, Regev's reduction has been used as a quantum algorithmic tool for providing a quantum advantage for variants of the decoding problem. Following this line of work, the authors of [JSW+24] have recently come up with a quantum algorithm called Decoded Quantum Interferometry that is able to solve in polynomial time several optimization problems. They study in particular the Optimal Polynomial Interpolation (OPI) problem, which can be seen as a decoding problem on Reed-Solomon codes. In this work, we provide strong improvements for some instantiations of the OPI problem. The most notable improvements are for the problem (originating from lattice-based cryptography) on Reed-Solomon codes but we also study different constraints for OPI. Our results provide natural and convincing decoding problems for which we believe to have a quantum advantage. Our proof techniques involve the use of a soft decoder for Reed-Solomon codes, namely the decoding algorithm from Koetter and Vardy [KV03]. In order to be able to use this decoder in the setting of Regev's reduction, we provide a novel generic reduction from a syndrome decoding problem to a coset sampling problem, providing a powerful and simple to use theorem, which generalizes previous work and is of independent interest. We also provide an extensive study of OPI using the Koetter and Vardy algorithm.

Paper Structure

This paper contains 44 sections, 31 theorems, 92 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Context
  3. Quantum advantage with "decoded quantum interferometry".
  4. Regev's reduction.
  5. The $\hbox{SIS}_{\infty}$ and $\hbox{ISIS}_{\infty}$ problems.
  6. Improving Regev's reduction by taking better decoders.
  7. The issue with errors in the decoder.
  8. Overview of our contributions
  9. Generic reduction theorem for the inhomogeneous setting
  10. Comparison with existing algorithms.
  11. Using the KV decoder for solving the $\cS\textrm{-Polynomial Interpolation}$ problem
  12. Comparison with existing work.
  13. How to read this plot.
  14. Other results.
  15. Preliminaries
  16. ...and 29 more sections

Key Result

Theorem 1

For a function $f : \mathbb{F}_q^n \rightarrow \mathbb{C}$, if we have access to an algorithm that efficiently solves $\mathrm{SD}(\mathbf{H},|f|^2)$ with some error $P_{dec} \ge \frac{1}{\poly(n)}$ then we can construct an efficient quantum algorithm for $\mathrm{CS}(\mathbf{H}^{\bot},|\widehat{f}|

Figures (3)

  • Figure 1: Minimum admissible degree ratio $R = \frac{k}{n}$ as a function of $\rho = \frac{|S|}{q}$. The (top) purple line corresponds to the Decoded Quantum Interferometry algorithm for $\cS\textrm{-Polynomial Interpolation}$. The (middle) green line corresponds to our algorithm for $\cS\textrm{-Polynomial Interpolation}$ with random $S$ and the (bottom) blue line corresponds to our algorithm with $S = \llbracket -u , u \rrbracket$.
  • Figure 2: Admissible rate $R' = \frac{k'}{n}$ as a function of $\rho = \frac{2u+1}{q}$. The blue line corresponds to the function described above and the red line corresponds to the uniform distribution. These are numerical results with $q = 1009$.
  • Figure 3: Minimum admissible rate $R = \frac{k}{n}$ as a function of $\rho = \frac{|S|}{q}$. The (top) purple line corresponds to the Decoded Quantum Interferometry algorithm for $\cS\textrm{-Polynomial Interpolation}$. The (middle) green line corresponds to our algorithm for $\cS\textrm{-Polynomial Interpolation}$ with random $S$ and the (bottom) blue line corresponds to our algorithm with $S = \llbracket -u , u \rrbracket$.

Theorems & Definitions (58)

  • Definition 1
  • Theorem 1: Informal
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Proposition 1: JSW+24
  • Definition 2
  • Proposition 2
  • Definition 3
  • Proposition 3
  • ...and 48 more