Provable learning of quantum states with graphical models

The complete learning of any unknown many-body systems requires a number of samples exponentially in the particle number. Several works consider subclasses of physical meaningful states such as stabilizer states or high-temperature Gibbs states that can be learned in polynomial sample complexity. Others consider to learn states under special assumptions, such as PAC learning and shadow tomography. In this work, we consider learning many-body systems that can be approximately represented by restricted Boltzmann machines (RBMs). We exhibit robustness results for efficient provable two-hop neighborhood learning method for ferromagnetic and locally consistent RBMs. We consider the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:msub> <a:mi>L</a:mi> <a:mi>p</a:mi> </a:msub> </a:math> norm as a distance measure, including both total variation distance and max-norm distance in the limit. We show that various quantum states can be learned with a sample complexity exponentially better than the naive state tomography. Our results establish a broader theoretical framework for the efficient learning of complex quantum many-body systems.