Approximating Gibbs states of local Hamiltonians efficiently with projected entangled pair 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(log22 (N/epsilon))) if beta < O (log(2) N). Second, building on the work of Hastings [M.B. Hastings, Phys. Rev. B 73, 085115 (2006)], we derive a polynomial scaling relation, D = (N/epsilon)(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(log2 N) whenever the density of states only grows polynomially in the system size. All results hold for any spatial dimension of the lattice

Approximating local observables on projected entangled pair states

“…It remains a very interesting problem for future work to extend our results to G-injective PEPS. What we have shown here can to an extent be seen as the ground state analogue of an insight that has already been established for high-temperature Gibbs states, for which both an efficient approximation in terms of tensor network states with polynomial bond dimension and an efficient computation of expectation values have both been identified [25,26]. We hence contribute to demystifying the complexity of contracting tensor network states, coming to the conclusion that the situation for higher dimensional systems is not that different compared to 1D systems, for which the DMRG approach provides the workhorse of numerical studies.…”

“…For systems with d > 1, the closest result to Conjecture 1 we are aware of is the one presented in ref. [25], which uses a specific assumption on the density of states to find D = e O(log 2 (N/ ) d+1 ) , which is quasi-polynomial in N/ for constant d. While this is perfectly reasonable, the conjecture misses the point that it does not necessarily capture key properties of the true ground state. Again for 1D systems, injectivity of the MPS will readily imply exponentially decaying correlations.…”

“…But the expectation that the physical corner is well approximated by PEPS, dubbed the "PEPS conjecture", is still very much in place: This expectation is usually taken for granted and constitutes the basis of an entire research field. Indeed, for higher temperatures, a variant of this conjecture is actually provably true [25,26].…”

“…But the expectation that the physical corner is well approximated by PEPS, dubbed the "PEPS conjecture", is still very much in place: This expectation is usually taken for granted and constitutes the basis of an entire research field. Indeed, for higher temperatures, a variant of this conjecture is actually provably true [25,26].Having said all this, a new obstacle emerges for higherdimensional systems; one that is often seen as a key obstacle, a make-or-break issue when it comes to numerically simulating strongly correlated models with PEPS: Even though PEPS are expected to provide good approximations, they are believed to be not efficiently contractible to compute expectation values of local observables. This is backed up by a proof in worst-case complexity, stating that the contraction of two-dimensional PEPS is #P-complete [27].…”

Variational tensor network renormalization in imaginary time: Benchmark results in the Hubbard model at finite temperature

“…In 1D they can be represented by MPS prepared by accurate simulation of imaginary time evolution 18,19 of a system with ancillas. A similar approach can be applied in 2D [20][21][22] -the PEPS manifold is a compact representation for Gibbs states 23 although quality of its results is inferior when compared with a method presented in this article. Alternatively, direct contraction of the 3D tensor network representing the partition function was proposed 24 , but, due to local tensor update, they are expected to converge slowly with increasing refinement parameter.…”