Rényi free energy and variational approximations to thermal states

Abstract

We propose the construction of thermodynamic ensembles that minimize the Rényi free energy, as an alternative to Gibbs states. For large systems, the local properties of these Rényi ensembles coincide with those of thermal equilibrium, and they can be used as approximations to thermal states. We provide algorithms to find tensor network approximations to the 2-Rényi ensemble. In particular, a matrix-product-state representation can be found by using gradient-based optimization on Riemannian manifolds, or via a non-linear evolution which yields the desired state as a fixed point. We analyze the performance of the algorithms and the properties of the ensembles on one-dimensional spin chains.

References in corpus (51)

• The density-matrix renormalization group in the age of matrix product states
• Efficient classical simulation of slightly entangled quantum computations
• A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States
• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Matrix Product Density Operators: Simulation of finite-T and dissipative systems
• An Area Law for One Dimensional Quantum Systems
• Unifying time evolution and optimization with matrix product states
• Matrix product states represent ground states faithfully
• Area laws in quantum systems: mutual information and correlations
• Mixed-state dynamics in one-dimensional quantum lattice systems: a time-dependent superoperator renormalization algorithm
• DMRG and periodic boundary conditions: a quantum information perspective
• Spectral Gap and Exponential Decay of Correlations
• Matrix product operator representations
• Theory of finite-entanglement scaling at one-dimensional quantum critical points
• Finite Temperature Density Matrix Renormalization using an enlarged Hilbert space
• Scaling of entanglement support for Matrix Product States
• Minimally Entangled Typical Thermal State Algorithms
• Neural-Network Approach to Dissipative Quantum Many-Body Dynamics
• Variational optimization algorithms for uniform matrix product states
• Density Matrix Renormalization Group Method for 2D Classical Models
• Minimally entangled typical quantum states at finite temperature
• Solving Gapped Hamiltonians Locally
• Variational Quantum Monte Carlo Method with a Neural-Network Ansatz for Open Quantum Systems
• Variational neural network ansatz for steady states in open quantum systems
• Variational Matrix Product Operators for the Steady State of Dissipative Quantum Systems
• Tangent-space methods for uniform matrix product states
• A positive tensor network approach for simulating open quantum many-body systems
• Constructing neural stationary states for open quantum many-body systems
• Density matrix renormalisation group for a quantum spin chain at non-zero temperature
• Approximating Gibbs states of local Hamiltonians efficiently with PEPS
• Matrix product states for critical spin chains: finite size scaling versus finite entanglement scaling
• Locality of temperature
• A Matrix-Product-Operator Approach to the Nonequilibrium Steady State of Driven-Dissipative Quantum Arrays
• Projected Entangled Pair States at Finite Temperature: Imaginary Time Evolution with Ancillas
• Thermodynamic versus statistical nonequivalence of ensembles for the mean-field Blume-Emery-Griffiths model
• Exponential Thermal Tensor Network Approach for Quantum Lattice Models
• Purifications of multipartite states: limitations and constructive methods
• A tensor network annealing algorithm for two-dimensional thermal states
• Spectra and eigenstates of spin chain Hamiltonians
• Riemannian optimization of isometric tensor networks
• Improved thermal area law and quasi-linear time algorithm for quantum Gibbs states
• Maximum Renyi entropy principle for systems with power--law Hamiltonian
• Operator content of real-space entanglement spectra at conformal critical points
• Gaussian quantum fluctuations in interacting many particle systems
• Series-Expansion Thermal Tensor Network Approach for Quantum Lattice Models
• Fundamental limitations in the purifications of tensor networks
• Efficient description of many-body systems with Matrix Product Density Operators
• Fermionic Projected Entangled Pair States at Finite Temperature
• Spectral densities and partition functions of modular quantum systems as derived from a central limit theorem
• A variational approach for many-body systems at finite temperature
• Computing energy density in one dimension

Cited by in corpus (7)

• Isometric tensor network representations of two-dimensional thermal states
• Variational Neural and Tensor Network Approximations of Thermal States
• Random Projection using Random Quantum Circuits
• Variational post-selection for ground states and thermal states simulation
• Provably Efficient Simulation of 1D Long-Range Interacting Systems at Any Temperature
• The statistical mechanics and machine learning of the $\alpha$-Rényi ensemble
• Natural gradient and parameter estimation for quantum Boltzmann machines