Provable learning of quantum states with graphical models
Liming Zhao Naixu Guo Ming-Xing Luo Patrick Rebentrost
The complete learning of an $n$-qubit quantum state requires samples
exponentially in $n$. Several works consider subclasses of quantum states that
can be learned in polynomial sample complexity such as stabilizer states or
high-temperature Gibbs states. Other works consider a weaker sense of learning,
such as PAC learning and shadow tomography. In this work, we consider learning
states that are close to neural network quantum states, which can efficiently
be represented by a graphical model called restricted Boltzmann machines
(RBMs). To this end, we exhibit robustness results for efficient provable
two-hop neighborhood learning algorithms for ferromagnetic and locally
consistent RBMs. We consider the $L_p$-norm as a measure of closeness,
including both total variation distance and max-norm distance in the limit. Our
results allow certain quantum states to be learned with a sample complexity
\textit{exponentially} better than naive tomography. We hence provide new
classes of efficiently learnable quantum states and apply new strategies to
learn them.