Provable learning of quantum states with graphical models
This work provides new classes of efficiently learnable quantum states for quantum computing, though it is incremental as it builds on prior subclasses and weaker learning methods.
The paper tackles the problem of learning quantum states with exponential sample complexity by focusing on states close to neural network quantum states represented by restricted Boltzmann machines (RBMs), achieving sample complexity exponentially better than naive tomography.
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.