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arXiv:2006.15608 · Physical review. B./Physical review. B

Combination of tensor network states and Green's function Monte Carlo

Mingpu Qin

Published Sep 24, 2020• 6 citations
arXivHTMLPDF

Abstract

We propose an approach to study the ground state of quantum many-body systems in which tensor network states, specifically projected entangled pair states (PEPSs), and Green's function Monte Carlo (GFMC) are combined. PEPSs, by design, encode the area law which governs the scaling of entanglement entropy in quantum systems with short-range interactions but are hindered by the high computational complexity scaling with bond dimension $D$. GFMC is a highly efficient method, but it usually suffers from the infamous negative sign problem which can be avoided by the fixed-node approximation in which a guiding wave function is utilized to modify the sampling process. The trade-off for the absence of the negative sign problem is the introduction of systematic error by the guiding wave function. In this work, we combine these two methods, PEPS and GFMC, to take advantage of both of them. PEPSs are very accurate variational wave functions, while at the same time, only contractions of the single-layer tensor network are necessary in GFMC, which reduces the cost substantially. Moreover, energy obtained in GFMC is guaranteed to be variational and lower than the variational energy of the guiding PEPS wave function. Benchmark results of the ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model on the square lattice are provided.

DOI
10.1103/physrevb.102.125143

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We propose an approach to study the ground state of quantum many-body systems in which tensor network states, specifically projected entangled pair states (PEPSs), and Green's function Monte Carlo (GFMC) are combined. PEPSs, by design, encode the area law which governs the scaling of entanglement entropy in quantum systems with short-range interactions but are hindered by the high computational complexity scaling with bond dimension DDD. GFMC is a highly efficient method, but it usually suffers from the infamous negative sign problem which can be avoided by the fixed-node approximation in which a guiding wave function is utilized to modify the sampling process. The trade-off for the absence of the negative sign problem is the introduction of systematic error by the guiding wave function. In this work, we combine these two methods, PEPS and GFMC, to take advantage of both of them. PEPSs are very accurate variational wave functions, while at the same time, only contractions of the single-layer tensor network are necessary in GFMC, which reduces the cost substantially. Moreover, energy obtained in GFMC is guaranteed to be variational and lower than the variational energy of the guiding PEPS wave function. Benchmark results of the J1\ensuremath−J2{J}_{1}\ensuremath{-}{J}_{2}J1​\ensuremath−J2​ Heisenberg model on the square lattice are provided.
The tensor network algorithm, a family of prevalent numerical methods for quantum many-body problems, aptly captures the entanglement properties intrinsic to quantum systems, enabling precise representation of quantum states. However, its computational cost is notably high, particularly in calculating physical observables like correlation functions. To surmount the computational challenge and enhance efficiency, we propose integrating the Green's function Monte Carlo (GFMC) method with the projected entangled pair state (PEPS) ansatz. This approach combines the high-efficiency characteristics of Monte Carlo with the sign-free nature of tensor network states and proves effective in addressing the computational bottleneck. To showcase its prowess, we apply this hybrid approach to investigate the antiferromagnetic J1-J2 Heisenberg model on the square lattice, a model notorious for its sign problem in quantum Monte Carlo simulations. Our results reveal a substantial improvement in the accuracy of ground-state energy when utilizing a preliminary PEPS as the guiding wave function for GFMC. By calculating the structure factor and spin-spin correlation functions, we further characterize the phase diagram, identifying a possible columnar valence-bond state phase within the intermediate parameter range of 0.52<J2/J1<0.58. This comprehensive study underscores the efficacy of our combined approach, demonstrating its ability to accurately simulate frustrated quantum spin systems while ensuring computational efficiency.
The tensor network algorithm, a family of prevalent numerical methods for quantum many-body problems, aptly captures the entanglement properties intrinsic to quantum systems, enabling precise representation of quantum states. However, its computational cost is notably high, particularly in calculating physical observables like correlation functions. To surmount the computational challenge and enhance efficiency, we propose integrating the Green's function Monte Carlo (GFMC) method with the projected entangled pair state (PEPS) ansatz. This approach combines the high-efficiency characteristics of Monte Carlo with the sign-free nature of tensor network states and proves effective in addressing the computational bottleneck. To showcase its prowess, we apply this hybrid approach to investigate the antiferromagnetic ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the square lattice, a model notorious for its sign problem in quantum Monte Carlo simulations. Our results reveal a substantial improvement in the accuracy of ground-state energy when utilizing a preliminary PEPS as the guiding wave function for GFMC. By calculating the structure factor and spin-spin correlation functions, we further characterize the phase diagram, identifying a possible columnar valence-bond state phase within the intermediate parameter range of 0.52<J2/J1<0.580.52<{J}_{2}/{J}_{1}<0.580.52<J2​/J1​<0.58. This comprehensive study underscores the efficacy of our combined approach, demonstrating its ability to accurately simulate frustrated quantum spin systems while ensuring computational efficiency.
Abstract This paper introduces a hybrid approach combining Green’s function Monte Carlo (GFMC) method with projected entangled pair state (PEPS) ansatz. This hybrid method regards PEPS as a trial state and a guiding wave function in GFMC. By leveraging PEPS’s proficiency in capturing quantum state entanglement and GFMC’s efficient parallel architecture, the hybrid method is well-suited for the accurate and efficient treatment of frustrated quantum spin systems. As a benchmark, we applied this approach to study the frustrated J 1–J 2 Heisenberg model on a square lattice with periodic boundary conditions (PBCs). Compared with other numerical methods, our approach integrating PEPS and GFMC shows competitive accuracy in the performance of ground-state energy. This paper provides systematic and comprehensive discussion of the approach of our previous work [Phys. Rev. B 109 235133 (2024)].
We propose an efficient numerical method, which combines the advantages of recently developed tensor-network based methods and standard trial wave functions, to study the ground-state properties of quantum many-body systems. In this approach, we apply a projector in the form of a tensor-product operator to an input wave function, such as a Jastrow-type or Hartree-Fock wave function, and optimize the tensor elements via variational Monte Carlo. The entanglement already contained in the input wave function can considerably reduce the bond dimensions compared to the regular tensor-product state representation. In particular, this allows us to also represent states that do not obey the area law of entanglement entropy. In addition, for fermionic systems, the fermion sign structure can be encoded in the input wave function. We show that the optimized states provide good approximations of the ground-state energy and correlation functions in the cases of two-dimensional bosonic and fermonic systems.
Tensor network states, and in particular projected entangled pair states (PEPS) have been a strong ansatz for the variational study of complicated quantum many-body systems, thanks to their built-in entanglement entropy area law. In this work, we use a special kind of PEPS---gauged Gaussian fermionic PEPS (GGFPEPS)---to find the ground state of 2+12+12+1 dimensional pure Z2{\mathbb{Z}}_{2}Z2​ lattice gauge theories for a wide range of coupling constants. We do so by combining PEPS methods with Monte-Carlo computations, allowing for efficient contraction of the PEPS and computation of correlation functions. Previously, such numerical computations involved the calculation of the Pfaffian of a matrix scaling with the system size, forming a severe bottleneck; in this work we show how to overcome this problem. This paves the way for applying the method we propose and benchmark here to other gauge groups, higher dimensions, and models with fermionic matter, in an efficient, sign-problem-free way.
Restricted Boltzmann machine (RBM) has been proposed as a powerful variational ansatz to represent the ground state of a given quantum many-body system. On the other hand, as a shallow neural network, it is found that the RBM is still hardly able to capture the characteristics of systems with large sizes or complicated interactions. In order to find a way out of the dilemma, here, we propose to adopt the Green’s function Monte Carlo (GFMC) method for which the RBM is used as a guiding wave function. To demonstrate the implementation and effectiveness of the proposal, we have applied the proposal to study the frustrated J 1–J 2 Heisenberg model on a square lattice, which is considered as a typical model with sign problem for quantum Monte Carlo simulations. The calculation results demonstrate that the GFMC method can significantly further reduce the relative error of the ground-state energy on the basis of the RBM variational results. This encourages to combine the GFMC method with other neural networks like convolutional neural networks for dealing with more models with sign problem in the future.