Coarse grained belief propagation for simulation of interacting quantum systems at all temperatures

Abstract

We continue our numerical study of quantum belief propagation initiated in [Phys. Rev. A, 77 (2008), p. 052318]. We demonstrate how the method can be expressed in terms of an effective thermal potential that materializes when the system presents quantum correlations, but is insensitive to classical correlations. The thermal potential provides an efficient means to assess the precision of belief propagation on graphs with no loops. We illustrate these concepts using the one-dimensional quantum Ising model and compare our results with exact solutions. We also use the method to study the transverse field quantum Ising spin glass for which we obtain a phase diagram that is largely in agreement with the one obtained in [arXiv:0706.4391] using a different approach. Finally, we introduce the coarse grained belief propagation (CGBP) algorithm to improve belief propagation at low temperatures. This method combines the reliability of belief propagation at high temperatures with the ability of entanglement renormalization to efficiently describe low energy subspaces of quantum systems with local interactions. With CGBP, thermodynamic properties of quantum systems can be calculated with a high degree of accuracy at all temperatures.

updated references and acknowledgements

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• Clustering of conditional mutual information for quantum Gibbs states above a threshold temperature
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• Improved thermal area law and quasi-linear time algorithm for quantum Gibbs states
• Stability of topologically invariant order parameters at finite temperature
• Perturbative analysis of topological entanglement entropy from conditional independence
• Localized Virtual Purification
• The Quantum Cluster Variational Method and the Phase Diagram of the quantum ferromagnetic $J_1$-$J_2$ model
• Determining the structure of real-space entanglement spectrum from approximate conditional independence
• Clustering of conditional mutual information and quantum Markov structure at arbitrary temperatures
• Maximum-Likelihood-Estimate Hamiltonian learning via efficient and robust quantum likelihood gradient
• Quantum Cluster Variational Method and Message Passing Algorithms Revisited
• Strong decay of correlations for Gibbs states in any dimension
• Gibbs state sampling via cluster expansions