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.
Highlights
A fundamental task in quantum computing is that of learning a description of an unknown quantum state ρ
This is formalized as the problem of quantum state tomography, where we are granted the ability to form multiple copies of ρ and take arbitrary measurements, and must learn a state σ that is close to ρ in trace distance
We introduce to the quantum setting a well-known tool for noise-resilient classical Probably Approximately Correct” (PAC) learning, the statistical query (SQ) model, and define the problem of Statistical Query (SQ)-learning quantum states
Summary
A fundamental task in quantum computing is that of learning a description of an unknown quantum state ρ. (These theorems are formally stated as Corollaries 4.7, 4.11 and 4.12 respectively.) Our results position the problem of learning stabilizer states as a quantum analogue of the important classical problem of learning parities.2 In both cases there are simple “algebraic” learning algorithms for the noiseless setting, and the problem seems to become intractable with even the simplest kinds of noise. The algorithm of Rocchetto [41] joins a small class of PAC algorithms that do not fall into the SQ model, and do not admit any straightforward algorithms in noisy settings In our view, this frames learning stabilizer states with noise as one of the more compelling problems on the frontier of learning quantum states with noise. This form of differential privacy has recently been studied by [11]
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