Advanced Mean Field Methods in Quantum Probabilistic Inference - PowerPoint PPT Presentation

Advanced Mean Field Methods in Quantum Probabilistic Inference Kazuyuki Tanaka Graduate School of Information Sciences, Tohoku University E-mail: kazu@smapip.is.tohoku.ac.jp URL: http://www.smapip.is.tohoku.ac.jp/~kazu/ STATPHYS23 (Genova, Italy) July, 2007 1

Probability Distribution and Density Matrix Probability Distribution: Summation over all the 2 N possible configurations ( ) ∑ ∑ ∑ L L , , , W a a a 1 2 N = = = 0 , 1 0 , 1 0 , 1 a a a 1 2 N Density Matrix: Diagonalization of 2 N x2 N matrix Computational Complexity of Exponential Order O ( e N ) July, 2007 STATPHYS23 (Genova, Italy) 2

Conventional Belief Propagation Fundamental Structure of Conventional Belief Propagation ( ) ( ) ( ) = , , , , P a a a w a a w a a 1 2 3 12 1 2 23 2 3 ⎛ ⎞ ⎛ ⎞ ( ) ( ) ( ) ( ) ⎟ ∑∑ ∑ ∑ ⎜ ⎟ ⎜ ⎟ = = , , , , P a P a a a w a a w a a ⎜ ⎟ ⎜ 2 2 1 2 3 12 1 2 23 2 3 ⎝ ⎠ ⎝ ⎠ a a a a 1 3 1 3 We cannot factorize it in the similar way. We cannot factorize it in the similar way. + ≠ 1 H H H H 2 3 exp( ) exp( ) exp( ) 12 23 12 23 − − H H tr exp( ) exp( ) 13 12 23 ≠ − − H H ( tr exp( ))( tr exp( )) 1 12 3 23 July, 2007 STATPHYS23 (Genova, Italy) 3

Density Matrix and Reduced Density Matrix ∑ ˆ ≡ H H 1 ( ) ≡ − ρ H ij exp ∈ ij B Z Reduced Density ≡ ≡ ρ ρ ρ ρ tr tr Matrix \ \ ij ij i i j i ≡ Reducibility ρ ρ tr \ Condition i i ij July, 2007 STATPHYS23 (Genova, Italy) 4

Reduced Density Matrix and Effective Fields B ⎛ ⎞ i i 1 ∑ ⎜ ⎟ ≅ ρ λ Z exp ⎜ ⎟ → i k i ⎝ ⎠ ∈ B k i i ⎛ ⎞ ⎜ ⎟ 1 ∑ ∑ ≅ − + ⊗ + ⊗ ρ H λ I I λ exp ⎜ ⎟ → → ij ij k i l j ⎜ ⎟ Z ∈ ∈ ij k B \ j l B \ i ⎝ ⎠ i j j All effective field i B i \ j B j \ i are matrices July, 2007 STATPHYS23 (Genova, Italy) 5

Belief Propagation for Quantum Statistical Systems Propagating Rule of Effective Fields ∑ = − λ λ → → j i k i ∈ k B \ j i ⎛ ⎞ ⎡ ⎤ ⎛ ⎞ ⎜ ⎟ Z ∑ ∑ ⎜ ⎟ + ⎢ − + ⊗ + ⊗ ⎥ H λ I I λ i log tr exp ⎜ ⎟ ⎜ ⎟ → → ij k i l j \i ⎢ ⎥ Z ⎝ ⎠ ⎣ ⎦ ⎝ ⎠ ∈ ∈ k B \ j l B \ i ij i j j i ≡ ρ ρ tr \ i i ij Output July, 2007 STATPHYS23 (Genova, Italy) 6

Graphical Model for Probabilistic Inference { } = = = L Pr , , , A a A a A a 1 1 2 2 8 8 ( ) = L , , , P a a a 1 2 8 = ( , , ) ( ) ( , ) W a a a W a ,a ,a W a a 568 5 6 8 346 3 4 5 67 6 7 Undirected × ( , ) ( , ) ( , ) W a a W a a W a a Graph 25 2 5 24 2 4 13 1 3 A A 1 2 W W 24 13 W A A 3 4 Directed 25 W Graph 346 A A 6 5 W W 67 568 A A 7 8 STATPHYS23 (Genova, Italy) July, 2007 7

A Quantum-Statistical Extension of Probabilistic Inference 8 ( ) r r r ∑ ∑ ∑ ∑ ≡ − − x = ˆ Η S H log a W a a h γ γ γ x i r γ ∈ = γ ∈ 1 B a i B ( ) r r r h ∑ ∑ ˆ ≡ − − x x H S log a W a a γ γ γ i − r 1 B A A ∈ γ i 1 2 a i ˆ H ˆ H 24 13 ⎛ ⎞ ⎛ ⎞ ˆ 0 1 1 0 H A A ⎜ ⎟ 3 4 ≡ ⎜ ⎟ ≡ x S I 25 ˆ H ⎜ ⎟ ⎜ ⎟ 346 ⎝ ⎠ 1 0 ⎝ ⎠ 0 1 A A 6 5 ˆ ˆ H H { } = 67 568 13 , 24 , 25 , 346 , 568 , 67 B A A 7 8 STATPHYS23 (Genova, Italy) July, 2007 8

Numerical Results for Quantum Belief Propagation ⎛ ⎞ 1 0 Exact Exact = ≡ ⎜ ⎟ z S 1 h ⎜ ⎟ x − = = z z ⎝ ⎠ 0 1 S S ρ tr 0 . 9029 ... 1 1 = = z z S S ρ tr 0 . 8272 ... Undirected A A 4 4 1 2 Graph ˆ H ˆ H 24 13 ˆ H A A 3 4 Quantum Belief Propagation Belief Propagation Quantum 25 ˆ H 346 = = z z S S ρ tr 0 . 9032 ... A A 6 5 1 1 ˆ ˆ H H 568 67 = = z z S S ρ tr 0 . 8379 ... A A 4 4 7 8 STATPHYS23 (Genova, Italy) July, 2007 9

Linear Response Theory 〈〈 z z 〉〉 S S : A 8 3 A 2 1 1 λ − λ H H ∫ z z ≡ λ S S ρ tr( ) e e d 8 3 A A 3 4 0 − 〈 z 〉〈 z 〉 S S 8 3 A A 6 5 = ∈ x Ω { | } ~ x i i z − z S ρ S ρ tr tr 3 3 = A A lim 7 8 → h 0 h z z 1 1 ~ ≡ Z − ≡ − + z ρ H ρ H S exp( ) exp( ) h ~ 8 z Z July, 2007 STATPHYS23 (Genova, Italy) 10

Numerical Results for Canonical Correlations = ⎛ ⎞ 1 1 0 Exact Exact h ≡ ⎜ ⎟ z x S ⎜ ⎟ − ⎝ ⎠ 0 1 z z 〈〈 〉〉 = S S : 0 . 0918 ... 8 3 Undirected Graph 〈〈 z z 〉〉 = S S : 0 . 1727 ... 8 4 A A 1 2 ˆ H ˆ H 24 13 Quantum Belief Propagation Belief Propagation Quantum ˆ H A A 3 4 25 ˆ H 〈〈 z z 〉〉 = 346 S S : 0 . 0740 ... 8 3 A A 6 5 ˆ ˆ H H 〈〈 z z 〉〉 = S S : 0 . 0815 ... 67 568 8 4 A A 7 8 STATPHYS23 (Genova, Italy) July, 2007 11

Summary An Extension to Quantum Belief Propagation It is based on Quantum CVM (Morita, JPSJ1957) Future Problem Statistical-Mechanical Approaches to Quantum Statistical Inferences July, 2007 STATPHYS23 (Genova, Italy) 12