# Learning with Errors is easy with quantum samples

## TL;DR

This paper demonstrates an efficient quantum learning algorithm for the Learning with Errors problem, revealing implications for cryptography and the security of LWE-based schemes against quantum attacks.

## Contribution

It introduces a polynomial-time quantum learning algorithm for LWE with cryptographic error distributions, highlighting potential cryptographic vulnerabilities.

## Key findings

- Quantum sample complexity for LWE is polynomial.
- The algorithm does not break existing LWE encryption schemes.
- Potential for classical samples to approximate quantum states in cryptography.

## Abstract

Learning with Errors is one of the fundamental problems in computational learning theory and has in the last years become the cornerstone of post-quantum cryptography. In this work, we study the quantum sample complexity of Learning with Errors and show that there exists an efficient quantum learning algorithm (with polynomial sample and time complexity) for the Learning with Errors problem where the error distribution is the one used in cryptography. While our quantum learning algorithm does not break the LWE-based encryption schemes proposed in the cryptography literature, it does have some interesting implications for cryptography: first, when building an LWE-based scheme, one needs to be careful about the access to the public-key generation algorithm that is given to the adversary; second, our algorithm shows a possible way for attacking LWE-based encryption by using classical samples to approximate the quantum sample state, since then using our quantum learning algorithm would solve LWE.