The learnability scaling of quantum states: restricted Boltzmann machines

TL;DR

This paper investigates how the complexity of using restricted Boltzmann machines to reconstruct quantum states scales with the number of qubits, revealing quadratic growth near criticality and the importance of over-parametrization.

Contribution

It provides empirical evidence on the scaling behavior of RBMs for quantum state reconstruction and highlights the role of over-parametrization in effective learning.

Findings

RBM parameters scale quadratically with qubits near criticality

Pruning reduces parameters while maintaining accuracy

Over-parametrization facilitates learning in quantum state modeling

Abstract

Generative modeling with machine learning has provided a new perspective on the data-driven task of reconstructing quantum states from a set of qubit measurements. As increasingly large experimental quantum devices are built in laboratories, the question of how these machine learning techniques scale with the number of qubits is becoming crucial. We empirically study the scaling of restricted Boltzmann machines (RBMs) applied to reconstruct ground-state wavefunctions of the one-dimensional transverse-field Ising model from projective measurement data. We define a learning criterion via a threshold on the relative error in the energy estimator of the machine. With this criterion, we observe that the number of RBM weight parameters required for accurate representation of the ground state in the worst case - near criticality - scales quadratically with the number of qubits. By pruning small parameters of the trained model, we find that the number of weights can be significantly reduced while still retaining an accurate reconstruction. This provides evidence that over-parametrization of the RBM is required to facilitate the learning process.