Provable learning of quantum states with graphical models

TL;DR

This paper demonstrates that certain quantum states represented by neural network models can be learned efficiently with provable guarantees, significantly reducing sample complexity compared to naive methods.

Contribution

It introduces robust algorithms for learning quantum states modeled by RBMs, achieving exponential improvements in sample complexity over traditional quantum tomography.

Findings

Efficient two-hop neighborhood learning algorithms for RBMs.

Quantum states close to RBMs can be learned with exponential sample complexity reduction.

Provides robustness results for learning algorithms in quantum state estimation.

Abstract

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.