Approximating Gibbs states of local Hamiltonians efficiently with PEPS

TL;DR

This paper investigates how well PEPS can approximate Gibbs states of local quantum Hamiltonians, providing bounds on the bond dimension needed based on system size, temperature, and error tolerance, demonstrating efficiency in representation.

Contribution

The paper introduces new bounds on PEPS bond dimension for approximating Gibbs states, extending previous work and showing efficient representation across dimensions.

Findings

Bond dimension D scales as e^{O(log^2(N/ε))} for certain temperatures.

Polynomial scaling D=(N/ε)^{O(β)} for Gibbs state approximation.

Ground states can be approximated with D=N^{O(log N)} under polynomial density of states.

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.