Constructing exact representations of quantum many-body systems with deep neural networks

doi:10.1038/s41467-018-07520-3

. 2018 Dec 14;9(1):5322.

doi: 10.1038/s41467-018-07520-3.

Constructing exact representations of quantum many-body systems with deep neural networks

Giuseppe Carleo^{1

2}, Yusuke Nomura³, Masatoshi Imada³

Affiliations

¹ Center for Computational Quantum Physics, Flatiron Institute, 162 5th Avenue, New York, NY, 10010, USA. [email protected].
² Institute for Theoretical Physics, ETH Zurich, Wolfgang-Pauli-Str. 27, 8093, Zurich, Switzerland. [email protected].
³ Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-8656, Japan.

PMID: 30552316
PMCID: PMC6294148
DOI: 10.1038/s41467-018-07520-3

Constructing exact representations of quantum many-body systems with deep neural networks

Giuseppe Carleo et al. Nat Commun. 2018.

. 2018 Dec 14;9(1):5322.

doi: 10.1038/s41467-018-07520-3.

Authors

Giuseppe Carleo^{1

2}, Yusuke Nomura³, Masatoshi Imada³

Affiliations

¹ Center for Computational Quantum Physics, Flatiron Institute, 162 5th Avenue, New York, NY, 10010, USA. [email protected].
² Institute for Theoretical Physics, ETH Zurich, Wolfgang-Pauli-Str. 27, 8093, Zurich, Switzerland. [email protected].
³ Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-8656, Japan.

PMID: 30552316
PMCID: PMC6294148
DOI: 10.1038/s41467-018-07520-3

Abstract

Obtaining accurate properties of many-body interacting quantum matter is a long-standing challenge in theoretical physics and chemistry, rooting into the complexity of the many-body wave-function. Classical representations of many-body states constitute a key tool for both analytical and numerical approaches to interacting quantum problems. Here, we introduce a technique to construct classical representations of many-body quantum systems based on artificial neural networks. Our constructions are based on the deep Boltzmann machine architecture, in which two layers of hidden neurons mediate quantum correlations. The approach reproduces the exact imaginary-time evolution for many-body lattice Hamiltonians, is completely deterministic, and yields networks with a polynomially-scaling number of neurons. We provide examples where physical properties of spin Hamiltonians can be efficiently obtained. Also, we show how systematic improvements upon existing restricted Boltzmann machines ansatze can be obtained. Our method is an alternative to the standard path integral and opens new routes in representing quantum many-body states.

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Conflict of interest statement

The authors declare no competing interests.

Figures

**Fig. 1**
Structure of deep Boltzmann machine. Dots, squares, and triangles represent physical degrees of freedom $(σ_{i}^{z})$ , hidden units (h_j), deep units (d_k), respectively. Solid curves represent interlayer couplings (W_ij and $W_{j k}^{'}$ )

**Fig. 2**
Construction of exact DBM representations of the transverse-field Ising model. In this example, a step of imaginary-time evolution is shown, for the case of the one-dimensional transverse-field Ising model. Dots represent physical degrees of freedom $(σ_{i}^{z})$ , squares represent hidden units (h_j), triangles represent deep units (d_k). In each panel, upper networks are the initial state with arbitrary network form, and the bottom networks are the final states, after application of the propagator. Intermediate steps illustrate how the network is modified, where the relevant modified couplings at each step are highlighted in black. The highlighted solid and dashed curves indicate new and vanishing couplings, respectively. a Shows the diagonal (interaction) propagator being applied to the highlighted blue spins. This introduces a hidden unit (green) connected only to the two physical spins. In (b) the off-diagonal (transverse-field) propagator is applied, acting on the blue physical spin. Here, we then add one deep unit (red triangle), and a hidden unit (green) mediating visible–deep interactions

**Fig. 3**
Construction of exact DBM representations of Heisenberg models. In this example, a time step of imaginary-time evolution is shown, for the case of the one-dimensional antiferromagnetic Heisenberg model. Dots represent physical degrees of freedom $(σ_{i}^{z})$ , squares represent hidden units (h_j), triangles represent deep units (d_k). The three panels (a–c) represent different possible explicit constructions. In each panel, upper networks are the initial state with arbitrary network form, and the bottom networks are the final states, after application of the propagator. Intermediate steps illustrate how the network is modified, where the relevant modified weights at each step are highlighted in black. In those diagrams, dashed lines indicate that the corresponding weights are set to zero, and dotted lines indicate complex-valued weights. The three panels correspond to the (a) “1 deep, 3 hidden” (1d–3h), (b) “2 deep, 6 hidden” (2d–6h), and (c) “2 deep, 4 hidden” (2d–4h) constructions (see text for a more detailed explanation of the individual steps characteristic of each construction)

**Fig. 4**
Imaginary-time evolution with a DBM for 1D spin models. a Expectation value of energy of the transverse-field Ising Hamiltonian in the exact imaginary-time evolution (continuous line) is compared to the stochastic result obtained with a DBM (filled circles) (δ_τ = 0.01). Empty circles correspond to the approximate RBM evolution scheme, Eq. (15). We consider the critical point (Γ_l = V_lm), periodic boundary conditions, and N = 20 sites. b Expectation value of the isotropic antiferromagnetic Heisenberg Hamiltonian (AFHM) in the exact imaginary-time evolution (continuous line) is compared to the stochastic result obtained with a DBM (δ_τ = 0.01) following the 2d–6h construction. We consider periodic boundary conditions, N = 16 sites. The subscript α in DBM_α in panels (a, b) specifies a different initial state $∣Ψ_{0}⟩$ : α = 1 means that the initial state is an RBM state with hidden-unit density M/N = 1, whereas when α = 0 the initial state is the empty-network state (M = 0). All energies are in units of the transverse field (Γ_l = 1) for the TFIM, and of the exchange (J = 1) for the AFHM

**Fig. 5**
Approaching the exact ground-state energy. a Relative error on the ground-state energy for the 1D AFHM as a function of the imaginary time. Here we consider periodic boundary conditions, N = 80 sites, and δ_τ = 0.01, in units of the exchange J = 1. The subscript α in DBM_α specifies a different initial state $∣Ψ_{0}⟩$ : α = 1 means that the initial state is an RBM state with hidden-unit density M/N = 1, whereas when α = 0 the initial state is the empty-network state (M = 0). b Relative error on the ground-state energy for the two-dimensional J₁ – J₂ AFHM as a function of the imaginary time. As an energy unit, we consider J₁ = 1, and take J₂ = 0.0 and 0.4, periodic boundary conditions, N = 4×4 = 16 sites, and δ_τ = 0.001. Initial states are pre-optimized pair-product (geminal) state $∣ψ_{PP}⟩$ supplemented by Gutzwiller factor $P_{G}^{\infty}$ = $\prod_{l} (1 - n_{l ↑} n_{l ↓})$ prohibiting double occupancy and quantum number projection onto the singlet state $L^{S = 0}$ , i.e., $∣Ψ_{0}⟩$ = $L^{S = 0} P_{G}^{\infty} ∣ψ_{PP}⟩$ . The PP states are given by $∣ψ_{PP}⟩$ = ${(\sum_{l, m = 1}^{N} f_{l m}^{↑ ↓} c_{l ↑}^{†} c_{m ↓}^{†})}^{N ∕ 2} ∣0⟩$ , where $f_{l m}^{↑ ↓}$ are variational parameters and $c_{l σ}^{†}$ are the operators creating the electron with spin σ at lth site

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References

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1. Abrikosov AA. Methods of Quantum Field Theory in Statistical Physics. New York: Dover Publications; 1975.

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[1] Feynman RP. Space-time approach to non-relativistic quantum mechanics. Rev. Mod. Phys. 1948;20:367–387. doi: 10.1103/RevModPhys.20.367. - DOI

[2] Feynman RP. Space-time approach to non-relativistic quantum mechanics. Rev. Mod. Phys. 1948;20:367–387. doi: 10.1103/RevModPhys.20.367. - DOI

[3] Dyson FJ. The S matrix in quantum electrodynamics. Phys. Rev. 1949;75:1736–1755. doi: 10.1103/PhysRev.75.1736. - DOI

[4] Dyson FJ. The S matrix in quantum electrodynamics. Phys. Rev. 1949;75:1736–1755. doi: 10.1103/PhysRev.75.1736. - DOI

[5] Hubbard J. Calculation of partition functions. Phys. Rev. Lett. 1959;3:77–78. doi: 10.1103/PhysRevLett.3.77. - DOI

[6] Hubbard J. Calculation of partition functions. Phys. Rev. Lett. 1959;3:77–78. doi: 10.1103/PhysRevLett.3.77. - DOI

[7] Stratonovich RL. On a method of calculating quantum distribution functions. Sov. Phys. Dokl. 1957;2:416–419.

[8] Stratonovich RL. On a method of calculating quantum distribution functions. Sov. Phys. Dokl. 1957;2:416–419.

[9] Abrikosov AA. Methods of Quantum Field Theory in Statistical Physics. New York: Dover Publications; 1975.

[10] Abrikosov AA. Methods of Quantum Field Theory in Statistical Physics. New York: Dover Publications; 1975.

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Constructing exact representations of quantum many-body systems with deep neural networks

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Constructing exact representations of quantum many-body systems with deep neural networks

Authors

Affiliations

Abstract

Conflict of interest statement

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