On the Hardness of PAC-learning Stabilizer States with Noise

TL;DR

This paper investigates the difficulty of learning stabilizer quantum states under noise within the PAC framework, establishing that the problem is computationally hard similar to classical Learning Parity with Noise, especially in noisy conditions.

Contribution

The paper introduces a quantum PAC-learning model with noise tolerance and proves exponential lower bounds, showing the problem's intractability akin to classical LPN.

Findings

Exponential lower bound on learning stabilizer states in the SQ model.

Learning stabilizer states with noise is as hard as classical Learning Parity with Noise.

The problem remains computationally hard even with simple noise models.

Abstract

We consider the problem of learning stabilizer states with noise in the Probably Approximately Correct (PAC) framework of Aaronson (2007) for learning quantum states. In the noiseless setting, an algorithm for this problem was recently given by Rocchetto (2018), but the noisy case was left open. Motivated by approaches to noise tolerance from classical learning theory, we introduce the Statistical Query (SQ) model for PAC-learning quantum states, and prove that algorithms in this model are indeed resilient to common forms of noise, including classification and depolarizing noise. We prove an exponential lower bound on learning stabilizer states in the SQ model. Even outside the SQ model, we prove that learning stabilizer states with noise is in general as hard as Learning Parity with Noise (LPN) using classical examples. Our results position the problem of learning stabilizer states as a natural quantum analogue of the classical problem of learning parities: easy in the noiseless setting, but seemingly intractable even with simple forms of noise.