Quantum many-body problems and neural...
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The University of Electro-Communications 電気通信大学 斎藤弘樹 Quantum many-body problems and neural networks
Transcript of Quantum many-body problems and neural...
The University of Electro-Communications
電気通信大学 斎藤弘樹
Quantum many-body problems and neural networks
The University of Electro-Communications
Application of machine learningto quantum mecanics
Science 355, 602 (2017)
Nat. Comm. 8, 662 (2017)
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Why neural network?
Science 355, 602 (2017)For example, in quantum spin systems,
N
| ⟩𝛹𝛹 = �𝑆𝑆
𝛹𝛹 𝑆𝑆 | ⟩𝑆𝑆
| ⟩𝑆𝑆 = | ⟩↑↓↑↓↓↑ ⋯ ↑↓↑
many-body quantum state
bases
number of bases ~ exp(N)
Size of Hilbert space explodes with N!
The University of Electro-Communications
Why neural network?
Science 355, 602 (2017)
| ⟩𝛹𝛹 = �𝑆𝑆
𝛹𝛹 𝑆𝑆 | ⟩𝑆𝑆
many-body quantum state
Neural network
Neural network is used as variational wave function.
Size of Hilbert space >> # of network parameters
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restricted Boltzmann machine
Visible
Hidden
Input
Output
weightshidden variablesbiases
input
How to use neural network?
Science 355, 602 (2017)
Network parameters are optimized to get desired
𝛹𝛹 𝑆𝑆
𝛹𝛹 𝑆𝑆
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Ground state
Minimize
𝐻𝐻 =∑𝑆𝑆𝑆𝑆′ Ψ∗ (𝑆𝑆) 𝑆𝑆 𝐻𝐻 𝑆𝑆′ Ψ(𝑆𝑆′)
∑𝑆𝑆 Ψ(𝑆𝑆) 2
Ψ(𝑆𝑆) 2Monte Carlo sampling of S with probability ∝
𝜕𝜕 𝐻𝐻𝜕𝜕𝑊𝑊
can be calculated similarly
Science 355, 602 (2017)
= �𝑆𝑆
𝑃𝑃(𝑆𝑆)�𝑆𝑆′𝑆𝑆′ 𝐻𝐻 𝑆𝑆
Ψ(𝑆𝑆′)Ψ(𝑆𝑆)
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Numerical results
Tranverse-field Ising model (TFI) Antiferromagnetic Heisenberg model (AFH)
TFI (1D 80sites) AFH (1D 80sites) AFH(2D 10x10)
erro
r
α ∝ number of hidden units
Science 355, 602 (2017)
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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 interaction
U J
U J
superfluid
insulator
phase transition
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Bases for Bose-Hubbard model
M sites, N particles
Number of bases satisfying
N M2 2 35 5 12610 10 92,37815 15 77,558,760
many-bodyquantum state
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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
1D 14 sites, 14 particles
Fully connected
14 40
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Hidden layer dependence1D 14 sites, 14 particles
Fully connected, single hidden layer two hidden layers
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Convolutional network
Fully connected
L : # of layersC : # of channelsF : filter size
Convolutional
CNN is better than fully connected network.
Translational symmetry
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Application to few-body problemsin continuous space
H. Saito, arXiv:1804.06521(to be published in JPSJ)
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Continuous space
𝛹𝛹(𝒙𝒙1,𝒙𝒙2,⋯ ,𝒙𝒙𝑁𝑁)many-body wave function
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 networkCOM coordinate is eliminated using Jacobi coordinate
Rerative distancesare input.
| |
| |
| |
Rotational degree of freedom is eliminated.
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Results
N = 3
N ≧ 3 Y. Yan and D. Blume, PRA 90, 013620 (2014)Path integral Monte Carlo
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Other works
Y. Nomura et al., PRB 96, 205152 (2017)
Fermi-Hubbard model
Time evolution of Ising model
S. Czischek et al., arXiv:1803.08321
G. Carleo et al., arXiv:1802.09558
Deterministic with deep Boltzmann
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SummaryThe neural network method [Carleo & Troyer, Science 355, 602 (2017)] was applied to
Deep convolutional network seems good.
Outlook Time evolution Other systems (frustration, random, …)
H. Saito, J. Phys. Soc. Jpn. 86, 093001 (2017)H. Saito and M. Kato, J. Phys. Soc. Jpn. 87, 014001 (2018)
H. Saito, arXiv:1804.06521
Bose-Hubbard model
Continuous-space problem
