Solving quantum many-body Hamiltonians with artificial neural - PowerPoint PPT Presentation

Solving quantum many-body Hamiltonians with artificial neural networks Yusuke NOMURA Univ. of Tokyo TNSAA 2018 2018/12/03 YN, A. Darmawan, Y. Yamaji, and M. Imada, PRB 96 , 205152 (2017) G. Carleo, YN, and M. Imada, arXiv:1802.09558, to appear in Nature Communications Collaborators: Andrew S. Darmawan, Youhei Yamaji, Giuseppe Carleo, Masatoshi Imada

Artificial Neural Networks in Condensed Matter Physics See also: H. Saito, JPSJ 86, 093001 (2017); L. Wang, PRE 96, 051301 (2017) L. Huang and L. Wang, PRB 95, 035105 (2017) Monte Carlo speed up (generative model) Many-body solver (generative model) YN et al., PRB 96, 205152 (2017) H. Saito and M. Kato, JPSJ 87, 014001 (2018); G. Carleo and M. Troyer Science 355, 602 (2017) J. Carrasquilla and R. G. Melko, Nat. Phys. 13, 431 (2017). Phase classification (discriminative model) N. Yoshioka et al., arXiv:1709.05790. T. Ohtsuki and T. Ohtsuki, JPSJ 86, 044708 (2017). A. Tanaka and A. Tomiya, JPSJ 86, 063001 (2017). T. Ohtsuki and T. Ohtsuki, JPSJ 85, 123706 (2016). E. P. L. van Nieuwenburg et al., Nat. Phys. 13, 435 (2017). (Related) G.Torlai and R. G. Melko PRB 94, 165134 (2016)

Restricted Boltzmann machine (RBM) Paul Smolensky (1986) G. E. Hinton, R. R. Salakhutdinov, Science. 313, 504 (2006) Single hidden layer + interlayer coupling only → restricted Boltzmann machine (RBM) Marginal distribution can represent any distribution over {0,1} N with infinite M Energy function Boltzmann distribution Marginal distribution K. Hornik, Neural Networks 4, 251 (1991); G. Cybenko, Mathematics of Control, Signals and Systems 2, 303 (1989); N. L. Roux and Y. Bengio, Neural Computation 20, 1631(2008). Mag. Field (bias term) b j Interaction W ij Mag. Field (bias term) a i

Using artificial neural network to solve quantum many body problems See also: H. Saito, JPSJ 86, 093001 (2017); H. Saito and M. Kato, arXiv:1709.05468 Quantum correlations among physical spins via artificial neural network Single hidden layer + interlayer coupling only → restricted Boltzmann machine (RBM) : real space spin config. : spin of hidden neuron G. Carleo and M. Troyer Science 355, 602 (2017) Variational wave function Mag. Field (bias term) b j | Ψ ( σ z ) | 2 = 1 X σ z ⇣X ⌘ X X X Ψ ( σ z ) = a i σ z σ z exp i + i W ij h j + b j h j { h j } i i,j j Interaction W ij σ z = σ z 1 , σ z 2 , . . . , σ z � � N h j = ± 1 Mag. Field (bias term) a i ⇣ ⌘ i a i σ z Y X P Ψ ( σ z ) = e W ij σ z 2 cosh b j + i × i j i

Optimization strategy many-body wave function = vector with exponentially large dimension → extract essential pattern from machine leaning and represent it with polynomial number of parameters → exact ψ(x) = teacher ( supervised learning ) according to ψ(x) → density estimation: estimate underlying probability density function ( unsupervised learning ) (most challenging) → finding unknown ground sate ※ For the moment, we will consider positive-definite wave function → wave function can be regarded as probability density function we teach machine “rule of the game” (commutation relation, Hamiltonian, measurement of energy, …) optimize parameters following variational principle = minimization of energy 1. We know exact ψ(x) 2. We do not know the form of ψ(x), but we can observe real-space configuration x generated 3. We do not know the form of ψ(x), we cannot observe real-space configuration x either

Example: 1D Antiferromagnetic Heisenberg model (8site) gauge transformation Initial Energy: wave function (real and positive for any x) E 0 : ground state energy Optimization following variational principle : σ x,y → -σ x,y for one of sublattice Optimized W ψ(x) Optimized RBM RBM interaction parameter Initial RBM x Exact h H i = h Ψ |H| Ψ i � E 0 h Ψ | Ψ i | Ψ ( x ) | 2 X h x |H| x 0 i Ψ ( x 0 ) h H i = p ( x ) E loc ( x ) p ( x ) = X E loc ( x ) = P x | Ψ ( x ) | 2 Ψ ( x ) x 0 x

G. Carleo and M. Troyer Science 355, 602 (2017) Using artificial neural network to solve quantum many body problems α: hidden variable density = (# hidden units)/(# physical spins) 1D Transverse-field Ising model 80 spins, periodic boundary condition h: transverse field (h=1: critical) 1D AF Heisenberg model 80 spins, periodic boundary condition 2D AF Heisenberg model 10x10 square lattice periodic boundary condition

Properties of RBM wave function → can be mapped onto string-bond state (= product of MPS) I. Glasser et al., PRX 8 , 011006 (2018); S. R. Clark, J. Phys. A: Math. Theor. 51 , 135301 (2018) D.-L. Deng et al., PRX 7 , 021021 (2017); J. Chen et al., PRB 97 , 085104 (2018) → volume-law entanglement entropy Y. Huang and J. E. Moore, arXiv:1701.06246 → area-law entanglement entropy → can be mapped onto entangled plaquette states (EPS) → k nonzero complex amplitudes ψ(x) can be represented by RBM with k hidden units 1. Complex RBM → can be applied to general wave function 2. Universal approximator 3. Short-range RBM P Y Ψ ( σ z ) = C p ( σ z p ) p =1 4. Long-range RBM Y � Y Ψ ( σ z ) = A i,j ( σ z Tr j ) i j ∈ i e b i /N + W ij σ z ✓ ◆ 0 j A i,j ( σ z j ) = e − b i /N − W ij σ z 0 j

RBM vs diagonal SBS I. Glasser et al., PRX 8 , 011006 (2018) J → 98 % overlap with Laughlin state in 4x4 lattice J=1, Jχ=1, square lattice, 10x10, open boundary RBM vs diagonal SBS RBM wave function Jχ e b i /N + W ij σ z ✓ ◆ Y � 0 j A i,j ( σ z Y Ψ ( σ z ) = A i,j ( σ z j ) = Tr j ) e − b i /N − W ij σ z 0 j i j ∈ i

NetKet: open-source package https://www.netket.org

1.Combine concepts from machine learning and physics 2. Adding additional hidden layer (deep Boltzmann machine) How to improve RBM wave function? G. Carleo, YN, and M. Imada, arXiv:1802.09558, to appear in Nat. Commun. YN, A. Darmawan, Y. Yamaji, and M. Imada, PRB 96, 205152 (2017) (3. Extension to fermion-boson coupled Hamiltonians)

1.Combine concepts from machine learning and physics 2. Adding additional hidden layer (deep Boltzmann machine) How to improve RBM wave function? YN, A. Darmawan, Y. Yamaji, and M. Imada, PRB 96, 205152 (2017) (3. Extension to fermion-boson coupled Hamiltonians) G. Carleo, YN, and M. Imada, arXiv:1802.09558, to appear in Nat. Commun.

Restricted Boltzmann machine (RBM) wave function Product-basis RBM (P-RBM) for quantum spins Fermi-sea-based RBM (F-RBM) for fermions Neural-network correlation factor : Product state: Neural-network correlation factor can be efficiently calculated because neuron spins are noninteracting G. Carleo and M. Troyer Science 355, 602 (2017) YN, A. Darmawan, Y. Yamaji, and M. Imada, PRB 96, 205152 (2017)

RBM+PP wave function ① restricted Boltzmann machine + pair-product combine concepts from machine learning (RBM) and physics (pair-product(PP) state) Product-basis RBM (P-RBM) RBM +PP YN, A. Darmawan, Y. Yamaji, and M. Imada, PRB 96, 205152 (2017) Pair-Product state (geminal wave function): no entanglement if hidden layer is absent direct entanglement in visible layer → help RBM to learn ground state

RBM+PP wave function ② restricted Boltzmann machine + pair-product RBM+PP wave function cf. many variable VMC wave function Gutzwiller Jastrow G. Carleo and M. Troyer Science 355, 602 (2017) D. Tahara and M. Imada JPSJ 77, 114701 (2008) neural-network correlation factor (number of visible variables) = N site (Heisenberg) = 2N site (Hubbard) Mag. Field (bias term) b j Interaction W ij Mag. Field (bias term) a i

Ability of RBM to represent Gutzwiller-Jastrow factor YN, A. Darmawan, Y. Yamaji, and M. Imada, PRB 96, 205152 (2017) Gutzwiller factor at site i rewrite except for constant factor and one-body potential RBM form except for constant factor W 2 W 1

Result for 2D Heisenberg model 2D Hubbard model Parameters are numerically optimized following variational principles: → finding set of parameters W which minimize energy using machine learning techniques stochastic reconfiguration method (condensed-matter physics community) natural gradient (artificial intelligence community) S. Sorella, PRB 64, 024512 (2001) S.-I. Amari, K. Kurata, and H. Nagaoka, IEEE Transactions on Neural Networks 3, 260 (1992) S.-I. Amari, Neural Comput. 10, 251 (1998).

Application to 2D antiferromagnetic Heisenberg model 8x8 square lattice with periodic boundary condition YN, A. Darmawan, Y. Yamaji, and M. Imada, PRB 96, 205152 (2017) tensor netwrok data: L. Wang et al., PRB 83,134421 (2011); F. Mezzacapo et al., New J. Phys. 11, 083026 (2009). tensor network (EPS and PEPS for virtual bond dim. 16) simple RBM RBM+PP (RVB) RBM+PP (RVB) + trans. sym. α = (# hidden units)/(# physical spins) RBM+PP substantially improves accuracy compared to P-RBM Improving reference function helps RBM

Application to 2D Hubbard model 8x8 square lattice, half-filling (periodic anti-periodic) YN, A. Darmawan, Y. Yamaji, and M. Imada, PRB 96, 205152 (2017) TNVMC data: H.-H. Zhao et al., PRB 96, 085103 (2017). α = (# hidden units)/(# physical spins) RBM+PP substantially improves accuracy compared to F-RBM Improving reference function helps RBM TNVMC

Application to 2D Hubbard model 8x8 square lattice, half-filling (periodic anti-periodic) YN, A. Darmawan, Y. Yamaji, and M. Imada, PRB 96, 205152 (2017) RBM+PP result for spin structure factor U dependence of RBM+PP energy (α=32) RBM+PP works better in larger U, in contrast to mVMC Heisenberg result