Approximating Gibbs states of local Hamiltonians efficiently with PEPS

Abstract

We analyze the error of approximating Gibbs states of local quantum spin Hamiltonians on lattices with Projected Entangled Pair States (PEPS) as a function of the bond dimension ($D$), temperature (${\beta }^{-1}$), and system size ($N$). First, we introduce a compression method in which the bond dimension scales as $D=e^{O({\log }^2(N/\epsilon ))}$ if $\beta <O(\log (N))$. Second, building on the work of Hastings [Phys. Rev. B 73, 085115 (2006)], we derive a polynomial scaling relation, $D={\left(N/\epsilon \right)}^{O(\beta )}$. This implies that the manifold of PEPS forms an efficient representation of Gibbs states of local quantum Hamiltonians. From those bounds it also follows that ground states can be approximated with $D=N^{O(\log (N))}$ whenever the density of states only grows polynomially in the system size. All results hold for any spatial dimension of the lattice.

12 pages, 1 figure

References in corpus (7)

• Matrix Product Density Operators: Simulation of finite-T and dissipative systems
• Matrix product states represent ground states faithfully
• Area laws in quantum systems: mutual information and correlations
• Entanglement entropy of fermions in any dimension and the Widom conjecture
• Valence Bond Solids for Quantum Computation
• Entropy and Entanglement in Quantum Ground States
• Approximating the ground state of gapped quantum spin systems

Cited by in corpus (54)

• Variational Matrix Product Operators for the Steady State of Dissipative Quantum Systems
• Operator Entanglement in Interacting Integrable Quantum Systems: the Case of the Rule 54 Chain
• Tensor Network Algorithms: a Route Map
• Quantum dynamics of thermalizing systems
• Variational tensor network renormalization in imaginary time: benchmark results in the Hubbard model at finite temperature
• Tensor network study of the $m=1/2$ magnetization plateau in the Shastry-Sutherland model at finite temperature
• Classical simulation of short-time quantum dynamics
• Time evolution of an infinite projected entangled pair state: a neighborhood tensor update
• Light cone tensor network and time evolution
• Quantum many-body systems in thermal equilibrium
• Exponential decay of mutual information for Gibbs states of local Hamiltonians
• Mixed s-sourcery: Building many-body states using bubbles of Nothing
• A universal formula for the entanglement asymmetry of matrix product states
• Computable Rényi mutual information: Area laws and correlations
• Efficient Large-Scale Many-Body Quantum Dynamics via Local-Information Time Evolution
• Time evolution of an infinite projected entangled pair state: a gradient tensor update in the tangent space
• One-dimensional quantum systems at finite temperatures can be simulated efficiently on classical computers
• Classical algorithms for many-body quantum systems at finite energies
• Certified algorithms for equilibrium states of local quantum Hamiltonians
• Relativistic continuous matrix product states for quantum fields without cutoff
• Simulation of many body localization and time crystals in two dimensions with the neighborhood tensor update
• Approximating the long time average of the density operator: Diagonal ensemble
• Benchmarking a novel efficient numerical method for localized 1D Fermi-Hubbard systems on a quantum simulator
• Efficient Representation of Minimally Entangled Typical Thermal States in two dimensions via Projected Entangled Pair States
• Mechanisms for the emergence of Gaussian correlations
• Exact projected entangled pair ground states with topological Euler invariant
• Thermal Area Law in Long-Range Interacting Systems
• Matrix product state approximations to quantum states of low energy variance
• Universal Spreading of Conditional Mutual Information in Noisy Random Circuits
• Open-system spin transport and operator weight dissipation in spin chains
• Tensor-network approach to thermalization in open quantum many-body systems
• A quantum information perspective on meson melting
• Dual-isometric Projected Entangled Pair States
• Locality optimization for parent Hamiltonians of Tensor Networks
• Localization crossover and subdiffusive transport in a classical facilitated network model of a disordered, interacting quantum spin chain
• When can a local Hamiltonian be recovered from a steady state?
• Low-temperature Gibbs states with tensor networks
• Locally accurate matrix product approximation to thermal states
• Universal contributions to charge fluctuations in spin chains at finite temperature
• Variational Neural and Tensor Network Approximations of Thermal States
• Large-$S$ and tensor-network methods for strongly-interacting topological insulators
• Simulating chiral spin liquids with projected entangled-pair states
• Tensor networks and efficient descriptions of classical data
• Dynamical quantum phase transition and thermal equilibrium in the lattice Thirring model
• Phases of Matrix Product States with Symmetric Quantum Circuits and Symmetric Measurements with Feedforward
• Bridging Rokhsar-Kivelson Type and Generic Quantum Phase Transitions via Thermofield Double States
• Provably Efficient Simulation of 1D Long-Range Interacting Systems at Any Temperature
• Conditional Independence of 1D Gibbs States with Applications to Efficient Learning
• Constructing k-local parent Lindbladians for matrix product density operators
• Parent Lindbladians for Matrix Product Density Operators
• Sample complexity of matrix product states at finite temperature
• High-temperature partition functions and classical simulatability of long-range quantum systems
• Efficient and simple Gibbs state preparation of the 2D toric code via duality to classical Ising chains
• Locally accurate tensor networks for thermal states and time evolution