Quantum many-body problems andneural networks

電気通信大学

斎藤弘樹

The University of Electro-Communications

Application of machine learning
to quantum mecanics

Science 355, 602 (2017)

Nat. Comm. 8, 662 (2017)

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Why neural network?
For example, in quantum spin systems,

many-body quantum state

Science 355, 602 (2017)|𝛹𝛹⟩ = � 𝛹𝛹 𝑆𝑆 |𝑆𝑆⟩
𝑆𝑆bases|𝑆𝑆⟩ = |↑↓↑↓↓↑ ⋯ ↑↓↑⟩
N

number of bases ~ exp(N)Size of Hilbert space explodes with N!

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Why neural network?

Science 355, 602 (2017)

many-body quantum state|𝛹𝛹⟩ = � 𝛹𝛹 𝑆𝑆 |𝑆𝑆⟩Neural network

𝑆𝑆

Neural network is used as variational wave function.
Size of Hilbert space >> # of network parameters

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How to use neural network?
restricted Boltzmann machine
HiddenScience 355, 602 (2017)

VisibleInput

Output𝛹𝛹 𝑆𝑆biases

weights

hiddenvariables

input

Network parameters are optimized to get desired 𝛹𝛹 𝑆𝑆
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Ground state

Science 355, 602 (2017)

Minimize∑𝑆𝑆𝑆𝑆 ′ Ψ ∗ (𝑆𝑆) 𝑆𝑆 𝐻𝐻 𝑆𝑆 ′ Ψ(𝑆𝑆 ′ )𝐻𝐻 =∑𝑆𝑆 Ψ(𝑆𝑆) 2= � 𝑃𝑃(𝑆𝑆) �𝑆𝑆𝑆𝑆 ′𝑆𝑆 ′ 𝐻𝐻 𝑆𝑆

Ψ(𝑆𝑆 ′ )Ψ(𝑆𝑆)Monte Carlo sampling of S with probability ∝𝜕𝜕 𝐻𝐻𝜕𝜕𝑊𝑊

Ψ(𝑆𝑆) 2can be calculated similarly

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Numerical results

Science 355, 602 (2017)

Tranverse-field Ising model (TFI)

AFH (1D 80sites)

AFH(2D 10x10)

error

TFI (1D 80sites)

Antiferromagnetic Heisenberg model (AFH)α ∝ number of hidden units

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Application to Bose-Hubbard model

H. Saito, J. Phys. Soc. Jpn. 86, 093001 (2017)
H. Saito and M. Kato, J. Phys. Soc. Jpn. 87, 014001 (2018)

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Bose-Hubbard model

hopping

UJ

interaction

superfluid

phase transitionUJ

insulator

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Bases for Bose-Hubbard model

many-bodyquantum state

M sites, N particles

Number of bases satisfying

N M
2 25 510 1015 153
12692,37877,558,760

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Neural networks used in this study
Feedforward network

1. Fully-connected network
2. Convolutional network
Activation function : tanh(x)Optimization scheme : Adam

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U dependence

Fully connected

1D 14 sites, 14 particles

1440

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Hidden layer dependence
1D 14 sites, 14 particles

Fully connected, single hidden layer

two hidden layers

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Convolutional network
Convolutional
Fully connected

CNN is better than fully
connected network.

Translational symmetry
L : # of layers
C : # of channels
F : filter size

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Application to few-body problems
in continuous space
H. Saito, arXiv:1804.06521(to be published in JPSJ)The University of Electro-Communications

Continuous space
many-body wave function𝛹𝛹(𝒙𝒙1 , 𝒙𝒙2 , ⋯ , 𝒙𝒙𝑁𝑁 )Fully connected network with single hidden layer

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Calogero-Sutherland model

harmonic potential

interaction

𝛽𝛽 = 2

Exact solution

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3D problem ---- Efimov trimer

harmonic potential

Gaussian attractive interaction

P. Naidon and S. Endo, Rep. Prog. Phys. 80, 056001 (2017)

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Input of network
COM coordinate is eliminated using Jacobi coordinate

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Rerative distances
are input.
Rotational degree of freedom is eliminated.

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Results

N=3N≧3Y. Yan and D. Blume, PRA 90, 013620 (2014)

Path integral Monte Carlo

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Other works
 Fermi-Hubbard model
Y. Nomura et al., PRB 96, 205152 (2017)

 Time evolution of Ising model
S. Czischek et al., arXiv:1803.08321

 Deterministic with deep Boltzmann
G. Carleo et al., arXiv:1802.09558

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Summary
The neural network method [Carleo & Troyer, Science 355, 602 (2017)] was applied to Bose-Hubbard model
Deep convolutional network seems good.
H. Saito, J. Phys. Soc. Jpn. 86, 093001 (2017)
H. Saito and M. Kato, J. Phys. Soc. Jpn. 87, 014001 (2018)

 Continuous-space problem

H. Saito, arXiv:1804.06521

Outlook Time evolution
 Other systems (frustration, random, …)

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