Entanglement transitions from restricted Boltzmann machines

Abstract

The search for novel entangled phases of matter has lead to the recent discovery of a new class of ``entanglement transitions'', exemplified by random tensor networks and monitored quantum circuits. Most known examples can be understood as some classical ordering transitions in an underlying statistical mechanics model, where entanglement maps onto the free energy cost of inserting a domain wall. In this paper, we study the possibility of entanglement transitions driven by physics beyond such statistical mechanics mappings. Motivated by recent applications of neural network-inspired variational Ansätze, we investigate under what conditions on the variational parameters these Ansätze can capture an entanglement transition. We study the entanglement scaling of short-range restricted Boltzmann machine (RBM) quantum states with random phases. For uncorrelated random phases, we analytically demonstrate the absence of an entanglement transition and reveal subtle finite size effects in finite size numerical simulations. Introducing phases with correlations decaying as $1/r^{\alpha }$ in real space, we observe three regions with a different scaling of entanglement entropy depending on the exponent $\alpha$. We study the nature of the transition between these regions, finding numerical evidence for critical behavior. Our work establishes the presence of long-range correlated phases in RBM-based wave functions as a required ingredient for entanglement transitions.

10 pages, 4 figures

References in corpus (9)

• The density-matrix renormalization group in the age of matrix product states
• Many body localization and thermalization in quantum statistical mechanics
• Learning phase transitions by confusion
• Critical properties of the measurement-induced transition in random quantum circuits
• Postselection-free entanglement dynamics via spacetime duality
• An exact mapping between the Variational Renormalization Group and Deep Learning
• A semi-classical limit for the many-body localization transition
• Deep learning and the renormalization group
• autoScale.py - A program for automatic finite-size scaling analyses: A user's guide

Cited by in corpus (1)

• Tensor networks for interpretable and efficient quantum-inspired machine learning