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Assessing the feasibility of quantum learning algorithms for noisy linear problems

Minkyu Kim, Panjin Kim

TL;DR

There exist efficient classical algorithms for short integer solution and size-reduced learning with errors problems if the quantum samples used by the previous studies are given.

Abstract

Quantum algorithms for solving noisy linear problems are reexamined, under the same assumptions taken from the existing literature. The findings of this work include on the one hand extended applicability of the quantum Fourier transform to the ring learning with errors problem which has been left open by Grilo et al., who first devised a polynomial-time quantum algorithm for solving noisy linear problems with quantum samples. On the other hand, this paper also shows there exist efficient classical algorithms for short integer solution and size-reduced learning with errors problems if the quantum samples used by the previous studies are given.

Assessing the feasibility of quantum learning algorithms for noisy linear problems

TL;DR

There exist efficient classical algorithms for short integer solution and size-reduced learning with errors problems if the quantum samples used by the previous studies are given.

Abstract

Quantum algorithms for solving noisy linear problems are reexamined, under the same assumptions taken from the existing literature. The findings of this work include on the one hand extended applicability of the quantum Fourier transform to the ring learning with errors problem which has been left open by Grilo et al., who first devised a polynomial-time quantum algorithm for solving noisy linear problems with quantum samples. On the other hand, this paper also shows there exist efficient classical algorithms for short integer solution and size-reduced learning with errors problems if the quantum samples used by the previous studies are given.
Paper Structure (15 sections, 2 theorems, 35 equations, 2 figures, 7 algorithms)

This paper contains 15 sections, 2 theorems, 35 equations, 2 figures, 7 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Revisiting previous works
  4. Conclusion
  5. Data availability
  6. Author contributions
  7. Additional information

Key Result

Lemma 1

For any $a(X), b(X) \in \mathcal{R}$, the following relations hold:

Figures (2)

  • Figure 1: (a) A circuit description of BV algorithm and (b) its variant
  • Figure 2: Success rate of the proposed algorithm as a function of the error bound for (a) $q=3329$ (Kyber) and for (b) $q=2^{23} -2^{13} + 1$ (Dilithium). In each figure, the condition given by Equation (\ref{['eq:LG-condition']}) is around 20 and 1023, respectively. In (b), the interval in the horizontal axis is 10.

Theorems & Definitions (4)

  • Lemma 1
  • proof
  • Proposition 1
  • proof