Approximating Gibbs states of local Hamiltonians efficiently with projected entangled pair states

2014/06/11 by András Molnár, Norbert Schuch, Frank Verstraete +2 · 5 citations
Computer Science · Mathematics · Physics and Astronomy · #Combinatorics #Dimension (graph theory) #Geometry #Lattice (music) #Mathematical physics #Mathematics #Physics #Quantum #Quantum Computing Algorithms and Architecture #Quantum and electron transport phenomena #Quantum many-body systems #Quantum mechanics #Scaling #cond-mat.stat-mech #cond-mat.str-el #quant-ph

paper · pdf · doi:10.1103/physrevb.91.045138

published as Phys. Rev. B 91, 045138 (2015) · 12 pages, 1 figure

arxiv created 2014/06/11 · openalex publication_date 2015/01/29 · arxiv updated 2015/02/16 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

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 (\ensuremathβ^\ensuremath-1), and system size (N). First, we introduce a compression method in which the bond dimension scales as D=e^O(log₂²(N/\ensuremathε)) if \ensuremathβ<O(log₂N). Second, building on the work of Hastings [M. B. Hastings, Phys. Rev. B 73, 085115 (2006)], we derive a polynomial scaling relation, D=(N/\ensuremathε)^O(\ensuremathβ). 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(log₂N) whenever the density of states only grows polynomially in the system size. All results hold for any spatial dimension of the lattice.

Citations

• Matrix Product Density Operators: Simulation of Finite-Temperature and Dissipative Systems
• Colloquium: Area laws for the entanglement entropy
• Solving gapped Hamiltonians locally
• Entropy and Area
• Efficient Quantum Algorithms for Simulating Sparse Hamiltonians
• Fermionic projected entangled pair states
• Matrix product states represent ground states faithfully
• Valence-bond states for quantum computation
• Matrix Analysis

Cited by

• Structure of matrix product locally purifiable density operators
• Normal projected entangled pair states generating the same state
• Extracting average properties of disordered spin chains with translationally invariant tensor networks
• Locality of temperature in spin chains
• Exponential decay of mutual information for Gibbs states of local Hamiltonians

Related

• PEPS as unique ground states of local Hamiltonians
• Approximating local properties by tensor network states with constant bond dimension
• Criticality, the area law, and the computational power of PEPS
• Local Hamiltonians with Approximation-Robust Entanglement
• Assessing the accuracy of projected entangled-pair states on infinite lattices