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arXiv:1406.2973 · Physical Review B

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

András Molnár, Norbert Schuch, Frank Verstraete, J. I. Cirac

Published Jan 29, 2015• 138 citations
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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 (${\ensuremath{\beta}}^{\ensuremath{-}1}$), and system size ($N$). First, we introduce a compression method in which the bond dimension scales as $D={e}^{O({log}_{2}^{2}(N/\ensuremath{\epsilon}))}$ if $\ensuremath{\beta}<O({log}_{2}N)$. Second, building on the work of Hastings [M. B. Hastings, Phys. Rev. B 73, 085115 (2006)], we derive a polynomial scaling relation, $D={(N/\ensuremath{\epsilon})}^{O(\ensuremath{\beta})}$. 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 $D={N}^{O({log}_{2}N)}$ whenever the density of states only grows polynomially in the system size. All results hold for any spatial dimension of the lattice.

DOI
10.1103/physrevb.91.045138

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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 (DDD), temperature (\ensuremathβ\ensuremath−1{\ensuremath{\beta}}^{\ensuremath{-}1}\ensuremathβ\ensuremath−1), and system size (NNN). First, we introduce a compression method in which the bond dimension scales as D=eO(log22(N/\ensuremathϵ))D={e}^{O({log}_{2}^{2}(N/\ensuremath{\epsilon}))}D=eO(log22​(N/\ensuremathϵ)) if \ensuremathβ<O(log2N)\ensuremath{\beta}<O({log}_{2}N)\ensuremathβ<O(log2​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/\ensuremathϵ)O(\ensuremathβ)D={(N/\ensuremath{\epsilon})}^{O(\ensuremath{\beta})}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=NO(log2N)D={N}^{O({log}_{2}N)}D=NO(log2​N) whenever the density of states only grows polynomially in the system size. All results hold for any spatial dimension of the lattice.
The minimally entangled typical thermal states (METTS) are an ensemble of pure states, equivalent to the Gibbs thermal state, designed with an efficient tensor network representation in mind. In this article, we use the projected entangled pair states (PEPS) as their representation on a two-dimensional (2D) lattice. Unlike matrix product states (MPS), which for 2D systems are limited by an exponential computational barrier in the lattice size, PEPS provides a more tractable approach. To substantiate the prowess of PEPS in modeling METTS (dubbed PEPS-METTS), we benchmark it against the purification method in the context of the 2D quantum Ising model at its critical temperature. Our analysis reveals that PEPS-METTS achieves accurate results with significantly lower bond dimensions. We further corroborate this finding in the 2D Fermi-Hubbard model. At a technical level, we introduce an efficient zipper method to obtain PEPS boundary MPS needed to compute expectation values and perform sampling. The imaginary time evolution is done with the neighborhood tensor update.
Projected entangled pair states (PEPS) methods have been proven to be powerful tools to solve strongly correlated quantum many-body problems in two dimensions. However, due to the high computational scaling with the virtual bond dimension DDD, in a practical application, PEPS are often limited to rather small bond dimensions, which may not be large enough for some highly entangled systems, for instance, frustrated systems. Optimization of the ground state using the imaginary time evolution method with a simple update scheme may go to a larger bond dimension. However, the accuracy of the rough approximation to the environment of the local tensors is questionable. Here, we demonstrate that by combining the imaginary time evolution method with a simple update, Monte Carlo sampling techniques and gradient optimization will offer an efficient method to calculate the PEPS ground state. By taking advantage of massive parallel computing, we can study quantum systems with larger bond dimensions up to D=10D=10D=10 without resorting to any symmetry. Benchmark tests of the method on the ${J}_{1}\ensuremath{-}{J}_{2}$ model give impressive accuracy compared with exact results.
We adapt and optimize the projected-pair-entangled-state (PEPS) algorithm on finite lattices (fPEPS) for two-dimensional Hubbard models and apply the algorithm to the Hubbard model with nearest-neighbor hopping on a square lattice. In particular, we formulate the PEPS algorithm using projected entangled pair operators, incorporate SU(2) symmetry in all tensor indices, and optimize the PEPS using both iterative-diagonalization-based local bond optimization and gradient-based optimization of the PEPS. We discuss the performance and convergence of the algorithm for the Hubbard model on lattice sizes of up to $8\ifmmode\times\else\texttimes\fi{}8$ for PEPS states with U(1) symmetric bond dimensions of up to D=8D=8D=8 and SU(2) symmetric bond dimensions of up to D=6D=6D=6. Finally, we comment on the relative and overall efficiency of schemes for optimizing fPEPS.
We introduce a family of states, the fermionic projected entangled pair states (fPEPS), which describe fermionic systems on lattices in arbitrary spatial dimensions. It constitutes the natural extension of another family of states, the PEPS, which efficiently approximate ground and thermal states of spin systems with short-range interactions. We give an explicit mapping between those families, which allows us to extend previous simulation methods to fermionic systems. We also show that fPEPS naturally arise as exact ground states of certain fermionic Hamiltonians. We give an example of such a Hamiltonian, exhibiting criticality while obeying an area law.
Projected entangled pair states (PEPS) offer memory-efficient representations of some quantum many-body states that obey an entanglement area law and are the basis for classical simulations of ground states in two-dimensional (2d) condensed matter systems. However, rigorous results show that exactly computing observables from a 2d PEPS state is generically a computationally hard problem. Yet approximation schemes for computing properties of 2d PEPS are regularly used, and empirically seen to succeed, for a large subclass of (``not too entangled'') condensed matter ground states. Adopting the philosophy of random matrix theory, in this work, we analyze the complexity of approximately contracting a 2d random PEPS by exploiting an analytic mapping to an effective replicated statistical mechanics model that permits a controlled analysis at a large bond dimension. Through this statistical-mechanics lens, we argue that (i) although approximately sampling wave-function amplitudes of random PEPS faces a computational-complexity phase transition above a critical bond dimension, and (ii) one can generically efficiently estimate the norm and correlation functions for any finite bond dimension. These results are supported numerically for various bond-dimension regimes. It is an important open question whether the above results for random PEPS apply more generally also to PEPS representing physically relevant ground states.