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.
翻译:我们在亚伦森(2007年)的“大概正确”框架内考虑学习稳定状态的问题,在亚伦森(2007年)的“大概正确”框架内,学习量子状态有噪音。在无噪音环境下,罗切托(2018年)最近给出了这个问题的算法,但这个吵闹的个案却被搁置。我们从古典学习理论的噪音容忍方法出发,为PAC学习量子状态引入了统计查询(SQ)模型,并证明这一模型的算法确实适应常见的噪音形式,包括分类和分解噪音。我们证明,在SQ模型中学习稳定状态的指数比SQ模型的指数要快得多。即便在SQ模型之外,我们也证明学习稳定状态与噪音平等一样困难。我们的结果将学习稳定状态的问题定位为典型学习等同的典型量子问题:在无噪音环境中很容易找到,但即使使用简单的噪音形式也看似棘手。