Deep learning-enhanced variational Monte Carlo method for quantum many-body physics

Yang, Li; Leng, Zhaoqi; Yu, Guangyuan; Patel, Ankit; Hu, Wen-Jun; Pu, Han

doi:10.1103/physrevresearch.2.012039

Cited by 40 publications

(28 citation statements)

References 48 publications

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“…This avoids the use of a finite basis set, a significant source of error for other Ansatz, and models higher-order electronelectron interactions compactly. The use of neural networks as a compact wave-function Ansatz has been studied before for spin systems [16][17][18][19][20] and many-electron systems on a lattice [19,21] as well as small systems of bosons in continuous space [22]. Applications of neural network Ansatz to chemical systems have been limited to date, presumably due to the complexity of Fermi-Dirac statistics.…”

Section: Introductionmentioning

confidence: 99%

Ab initio solution of the many-electron Schrödinger equation with deep neural networks

Pfau

Spencer

Matthews

et al. 2020

Phys. Rev. Research

389

331

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Given access to accurate solutions of the many-electron Schrödinger equation, nearly all chemistry could be derived from first principles. Exact wave functions of interesting chemical systems are out of reach because they are NP-hard to compute in general, but approximations can be found using polynomially scaling algorithms. The key challenge for many of these algorithms is the choice of wave function approximation, or Ansatz, which must trade off between efficiency and accuracy. Neural networks have shown impressive power as accurate practical function approximators and promise as a compact wave-function Ansatz for spin systems, but problems in electronic structure require wave functions that obey Fermi-Dirac statistics. Here we introduce a novel deep learning architecture, the Fermionic neural network, as a powerful wave-function Ansatz for many-electron systems. The Fermionic neural network is able to achieve accuracy beyond other variational quantum Monte Carlo Ansatz on a variety of atoms and small molecules. Using no data other than atomic positions and charges, we predict the dissociation curves of the nitrogen molecule and hydrogen chain, two challenging strongly correlated systems, to significantly higher accuracy than the coupled cluster method, widely considered the most accurate scalable method for quantum chemistry at equilibrium geometry. This demonstrates that deep neural networks can improve the accuracy of variational quantum Monte Carlo to the point where it outperforms other ab initio quantum chemistry methods, opening the possibility of accurate direct optimization of wave functions for previously intractable many-electron systems.

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Section: Introductionmentioning

confidence: 99%

Ab initio solution of the many-electron Schrödinger equation with deep neural networks

Pfau

Spencer

Matthews

et al. 2020

Phys. Rev. Research

389

331

View full text Add to dashboard Cite

show abstract

“…Deep learning techniques have recently impacted ab initio quantum chemistry by providing a new approach to the problem of tractable parameterization of high dimensional function spaces in quantum many-body problems. Over the past few years, a growing number of works [1][2][3][4][5][6][7][8][9][10][11] have demonstrated the use of neural networks in wavefunction approximation, with an increasing amount of importance placed on building symmetry constraints into models. In particular, several works [5,6,8,9,11] have recently applied neural networks to model antisymmetric wavefunctions.…”

Section: Introductionmentioning

confidence: 99%

Explicitly antisymmetrized neural network layers for variational Monte Carlo simulation

Lin¹,

Goldshlager²,

Lin³

2021

Preprint

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The combination of neural networks and quantum Monte Carlo methods has arisen as a promising path forward for highly accurate electronic structure calculations. Previous proposals have combined equivariant neural network layers with a final antisymmetric layer in order to satisfy the antisymmetry requirements of the electronic wavefunction. However, to date it is unclear if one can represent antisymmetric functions of physical interest, and it is difficult to precisely measure the expressiveness of the antisymmetric layer. This work attempts to address this problem by introducing explicitly antisymmetrized universal neural network layers. This approach has a computational cost which increases factorially with respect to the system size, but we are nonetheless able to apply it to small systems to better understand how the structure of the antisymmetric layer affects its performance. We first introduce a generic antisymmetric (GA) neural network layer, which we use to replace the entire antisymmetric layer of the highly accurate ansatz known as the FermiNet. We demonstrate that the resulting FermiNet-GA architecture can yield effectively the exact ground state energy for small atoms and molecules. We then consider a factorized antisymmetric (FA) layer which more directly generalizes the FermiNet by replacing the products of determinants with products of antisymmetrized neural networks. We find, interestingly, that the resulting FermiNet-FA architecture does not outperform the FermiNet. This strongly suggests that the sum of products of antisymmetries is a key limiting aspect of the FermiNet architecture. To explore this further, we investigate a slight modification of the FermiNet, called the full determinant mode, which replaces each product of determinants with a single combined determinant. We find that the full single-determinant FermiNet closes a large part of the gap between the standard single-determinant FermiNet and FermiNet-GA on small atomic and molecular problems. Surprisingly, on the nitrogen molecule at a dissociating bond length of 4.0 Bohr, the full single-determinant FermiNet can significantly outperform the largest standard FermiNet calculation with 64 determinants, yielding an energy within 0.4 kcal/mol of the best available computational benchmark.

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“…[39] In 1D systems matrix product state (MPS) based methods [40][41][42][43], like the DMRG approach [44][45][46] are very successful, even with periodic boundary conditions [47,48] and long range interactions, due to the area law entanglement [49][50][51], while for higher dimensions tensor network state approaches can be applied [52][53][54][55][56]. The model does not posses a sign problem [57][58][59] for unfrustrated bipartite lattices [33,60] and thus quantum Monte Carlo and more recent neural networkbased approaches [86][87][88][89][90][91][92][93][94][95][96] are highly effective in providing very accurate numerical solutions in higher dimensions.…”

Section: The Heisenberg Modelmentioning

confidence: 99%

Combined unitary and symmetric group approach applied to low-dimensional spin systems

Dobrautz,

Katukuri,

Bogdanov

et al. 2021

Preprint

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A novel combined unitary and symmetric group approach is used to study the spin-1 2 Heisenberg model and related Fermionic systems in a spin-adapted representation, using a linearlyparameterised Ansatz for the many-body wave function. We show that a more compact ground state wave function representation is obtained when combining the symmetric group, Sn, in the form of permutations of the underlying lattice site ordering, with the cumulative spin-coupling based on the unitary group, U(n). In one-dimensional systems the observed compression of the wave function is reminiscent of block-spin renormalization group approaches, and allows us to study larger lattices (here taken up to 80 sites) with the spin-adapted full configuration interaction quantum Monte Carlo method, which benefits from the sparsity of the Hamiltonian matrix and the corresponding sampled eigenstates that emerge from the reordering. We find that in an optimal lattice ordering the configuration state function with highest weight already captures with high accuracy the spin-spin correlation function of the exact ground state wave function. This feature is found for more general lattice models, such as the Hubbard model, and ab initio quantum chemical models, in this work exemplified by a one-dimensional hydrogen chain. We also provide numerical evidence that the optimal lattice ordering for the unitary group approach is not generally equivalent to the optimal ordering obtained for methods based on matrix-product states, such as the density-matrix renormalization group approach.

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Deep learning-enhanced variational Monte Carlo method for quantum many-body physics

Cited by 40 publications

References 48 publications

Ab initio solution of the many-electron Schrödinger equation with deep neural networks

Ab initio solution of the many-electron Schrödinger equation with deep neural networks

Explicitly antisymmetrized neural network layers for variational Monte Carlo simulation

Combined unitary and symmetric group approach applied to low-dimensional spin systems

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