Quantum many-body systems in thermal equilibrium

Abstract

The thermal or equilibrium ensemble is one of the most ubiquitous states of matter. For models comprised of many locally interacting quantum particles, it describes a wide range of physical situations, relevant to condensed matter physics, high energy physics, quantum chemistry and quantum computing, among others. We give a pedagogical overview of some of the most important universal features about the physics and complexity of these states, which have the locality of the Hamiltonian at its core. We focus on mathematically rigorous statements, many of them inspired by ideas and tools from quantum information theory. These include bounds on their correlations, the form of the subsystems, various statistical properties, and the performance of classical and quantum algorithms. We also include a summary of a few of the most important technical tools, as well as some self-contained proofs.

32 Pages 9 Figures. Updated version with updated sections and new proofs. Parts of these notes were the basis for a lecture series within the "Quantum Thermodynamics Summer School 2021" during August 2021 in Les Diablerets, Switzerland

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• From decay of correlations to locality and stability of the Gibbs state
• Structure of the Hamiltonian of mean force
• Estimating molecular thermal averages with the quantum equation of motion and informationally complete measurements
• Low-temperature Gibbs states with tensor networks
• Gibbs Sampling gives Quantum Advantage at Constant Temperatures with O(1)-Local Hamiltonians
• Quantum thermalization must occur in translation-invariant systems at high temperature
• Universality in the tripartite information after global quenches: spin flip and semilocal charges
• Strong decay of correlations for Gibbs states in any dimension
• Quantum concentration inequalities and equivalence of the thermodynamical ensembles: an optimal mass transport approach
• An Empirical Study of Quantum Dynamics as a Ground State Problem with Neural Quantum States
• Learning the structure of any Hamiltonian from minimal assumptions
• A Faster Algorithm for the Free Energy in One-Dimensional Quantum Systems
• High-temperature partition functions and classical simulatability of long-range quantum systems
• Entanglement structure for finite system under dual-unitary dynamics
• Constrained free energy minimization for the design of thermal states and stabilizer thermodynamic systems
• Efficient and simple Gibbs state preparation of the 2D toric code via duality to classical Ising chains
• Natural gradient and parameter estimation for quantum Boltzmann machines