Accelerated Monte Carlo simulations with restricted Boltzmann machines

Transcription

1 Accelerated simulations with restricted Boltzmann machines Li Huang 1 Lei Wang 2 1 China Academy of Engineering Physics 2 Institute of Physics, Chinese Academy of Sciences July 6, 2017 Machine Learning and Many-Body Physics, KITS, Beijing

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3 Quantum zoo 1 Auxiliary Field Quantum Variational Determinant Quantum World Line Quantum Continuous Time Quantum Diffusion Reptation Hybrid Quantum Gaussian Quantum Path Integral Diagrammatic Quantum Hisch-Fye Quantum Figure: Various quantum algorithms Ref: Quantum s, 2016

4 Quantum zoo 2 CT-HYB CT-INT LCT-INT CT-QMC CT-AUX CT-J Figure: Various continuous time quantum algorithms Ref: Rev Mod Phys 83, 349 (2011)

5 Troubles In my opinions, the applications of quantum methods are mainly hampered by the following three problems: How to visit the configuration space efficiently? Critical slowing down Negative sign problem (for fermionic system)

6 + Many-body physics 1 Detecting phase transitions in classic or quantum models Figure: Detecting phase transitions in Ising model with ANN Ref: Nat Phys 13, 431 (2017)

7 + Many-body physics 2 Representation of quantum many-body states Figure: Artificial neural network encoding a many-body quantum state Ref: Science 355, 602 (2017)

8 + Many-body physics 3 Solve the inverse problems Figure: Sparse modeling approach to analytical continuation of G(τ) Ref: Phys Rev E 95, (2017)

9 1 Is it possible to use ML tools to accelerate QMC? Supposed that it is difficult (not easy) to sample the target distributions by directly With the available data, we train a restricted Boltzmann machine () which can be viewed as a proxy (approximation) of the statistical distributions We simulate the trained, and then use it to propose new (local or non-local) updates The role played by the in the simulation is just like a recommender system in e-commerce platforms

10 2 How to visit the configuration space efficiently? The acceptance ratio is increased Critical slowing down The autocorrelation time is reduced Negative sign problem (for fermionic system) NO SOLUTION!

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12 1: Hamiltonian Hamiltonian: Ĥ FK = i,j Probability distribution: Free energy : ĉ i K ijĉ j + U N i=1 ( ˆn i 1 ) ( x i 1 ) 2 2 (1) p FK (x) = e FFK(x) /Z FK (2) F FK (x) = βu 2 N x i + ln det ( 1 + e βh) (3) i=1 x i {0, 1}: occupation number of the localized fermion at site i ˆn i ĉ i ĉi: occupation number operator of the mobile fermion K is the kinetic energy matrix of the mobile fermions

13 2: Phase diagram Ref: Phys Rev Lett 117, (2016) CDW: charge density wave insulator AI: Anderson insulator MI: Mott insulator FG: Fermi gas WL: weakly localized

14 3: Lattice QMC Bit-flip update: Detailed balance condition: x i 1 x i (4) Cons: T (x x ) A(x x ) T (x x) A(x x) = p FK(x ) p FK (x) (5) Scaling: O(N 4 ) Long autocorrelation times

15 1: Architecture The is a classical statistical mechanics system defined by the following energy function N M N M E (x, h) = a i x i b j h j x i W ij h j, (6) i=1 j=1 i=1 j=1 hidden layer h 1 h 2 h 3 h 4 h 5 h M b j W ij x 1 x 2 x 3 x N a i visible layer

16 2: Recent applications Connecting with tensor networks states Ψ (v) = h e E(v,h) = i,j e a iv i (1 + e b j+ i v iw ij ) (7) Ref: arxiv: Ψ MPS (v) = Tr ( nv ) A (i) [v i ] i=1 (8)

17 3: Recent applications Searching new cluster updates Ref: arxiv:

18 1: Collecting data Collecting training data set {x} by traditional method in a preliminary calculation

19 2: Training Joint probability distribution of the visible and hidden variables p(x, h) = e E(x,h) /Z (9) Marginal distribution p(x) = h p(x, h) = e F (x) /Z (10) Free energy of N M ) F (x) = a i x i + ln (1 + e bj+ N i=1 xiwij i=1 j=1 (11)

20 3: Training (cont) Free energy of N M ) F (x) = a i x i + ln (1 + e bj+ N i=1 xiwij i=1 j=1 (12) Free energy of F FK (x) = βu N x i + ln det ( 1 + e βh), (13) 2 i=1 We use F (x) to approximate F FK (x), and setup the parameters a i, b i, and W ij of

21 4: Training (cont) F(x) FK Test samples Figure: Fitting of the log probability of the to the one of the Falicov-Kamball model

22 5: Training (cont) Figure: Connection weights W ij of the (left) T/t = 015 (right) T/t = 013

23 6: Simulating We simulate the trained using the standard blocked Gibbs sampling approach The conditional probabilities of the hidden variables and visible variables read: p(h x) = p(x h) = p(h j = 1 x) = σ ( M p(h j x) (14) j=1 N p(x i h) (15) i=1 b j + ) N x i W ij i=1 j=1 (16) M p(x i = 1 h) = σ a i + W ij h j (17)

24 7: Simulating (cont) Std New h h apple p(h x) p(x 0 h) min 1, p(h0 ) p(h x) p(x 0 h 0 ) p(h) h 0 x x 0 x x 0 Figure: Two strategies of proposing updates using the

25 8: + MC We use the trained to propose new updates (instead of local bit-flip updates), and then apply the Metropolis algorithm to update the Markov chain [ A(x x ) = min 1, p(x) p(x ) pfk(x ] ) (18) p FK (x) Ideally, the acceptance ratio is one if the fits the Falicov- Kimball model perfectly

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27 Preliminary results Acceptance ratio (a) Local (G) (G+M) Autocorrelation time T/t (b) T/t Figure: (a) The acceptance ratio and (b) the total energy autocorrelation time of the

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29 1 Why does it work? The trained captures the target distribution correctly It is more efficient to explore the configuration space non-local updates vs local updates

30 2 Is this method general? No! Since the limitation of architecture, the MC algorithm must have binary degree of freedoms Ising and Z 2 gauge fields models Determinant quantum CT-AUX HF-QMC etc

31 3 Is this idea general? Sure We can consider as an efficient recommender system to accelerate simulations Similar works: Using classic gas model to accelerate CT-INT Ref: Phys Rev E 95, (2017) Using effective Ising model to accelerate DQMC and CT-AUX Ref: Phys Rev B 95, (2017), Phys Rev B 95, (2017), arxiv: , arxiv:

32 4 Is there space to improve it? Sure Self-guiding or online training approach Deep Boltzmann machines or deep belief networks

33 Take-home messages It is possible to design special recommender systems to accelerate some simulations The is a good recommender system for the simulation for Ref: Phys Rev B 95, (2017) Future works: Exploring reliable and efficient recommender systems for CT- HYB, which is the core computational engine in dynamical mean-field theory Ref: Phys Rev E 95, (2017)