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

July, 2007 STATPHYS23 (Genova, Italy) 1

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/

July, 2007 STATPHYS23 (Genova, Italy) 2

Probability Distribution: Summation

• ver all the 2N possible configurations

( )

∑ ∑ ∑

= = = 1 , 1 , 1 , 2 1

1 2

, , ,

a a a N

N

a a a W L L

Probability Distribution and Density Matrix

Density Matrix: Diagonalization

• f 2Nx2N matrix

Computational Complexity of Exponential Order O(eN)

July, 2007 STATPHYS23 (Genova, Italy) 3

Conventional Belief Propagation

( ) ( ) ( )

3 2 23 2 1 12 3 2 1

, , , , a a w a a w a a a P =

( ) ( ) ( ) ( )⎟

⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = =

∑ ∑ ∑∑

3 1 1 3

3 2 23 2 1 12 3 2 1 2 2

, , , ,

a a a a

a a w a a w a a a P a P

Fundamental Structure of Conventional Belief Propagation We cannot factorize it in the similar way. We cannot factorize it in the similar way. ) exp( ) exp( ) exp(

23 12 23 12

H H H H ≠ +

)) exp( tr ))( exp( tr ( ) exp( ) exp( tr

23 3 12 1 23 12 13

H H H H − − ≠ − −

1 2 3

July, 2007 STATPHYS23 (Genova, Italy) 4

Density Matrix and Reduced Density Matrix

∑

∈

≡

B ij ij

H H ˆ

( )

H ρ − ≡ exp 1 Z

ρ ρ

i i \

tr ≡

ρ ρ

ij ij \

tr ≡

ij i i

ρ ρ

\

tr ≡

Reduced Density Matrix Reducibility Condition

j

i

July, 2007 STATPHYS23 (Genova, Italy) 5

Reduced Density Matrix and Effective Fields

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ≅

∑

∈ →

i

B i k i

λ ρ

k i

Z exp 1

⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⊗ + ⊗ + − ≅

∑ ∑

∈ → ∈ → i \ B l j l j \ B k i k ij ij

j i

λ I I λ H ρ exp 1

ij

Z

i

j

i

All effective field are matrices

i

B j Bi \ i B j \

July, 2007 STATPHYS23 (Genova, Italy) 6

Belief Propagation for Quantum Statistical Systems

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⊗ + ⊗ + − + − =

∑ ∑ ∑

∈ → ∈ → ∈ → → i \ B l j l j \ B k i k ij j \ B k i k i j

j i i

λ I I λ H λ λ exp tr log

\i ij i

Z Z

Propagating Rule of Effective Fields

ij i i

ρ ρ

\

tr ≡

j

i

Output

July, 2007 STATPHYS23 (Genova, Italy) 7

Graphical Model for Probabilistic Inference

{ } ( )

) , ( ) , ( ) , ( ) , ( ) ( ) , , ( , , , , , , Pr

3 1 13 4 2 24 5 2 25 7 6 67 5 4 3 346 8 6 5 568 8 2 1 8 8 2 2 1 1

a a W a a W a a W a a W ,a ,a a W a a a W a a a P a A a A a A × = = = = = L L

Directed Graph

1

A

3

A

2

A

4

A

6

A

5

A

13

W

67

W

24

W

25

W

346

W

568

W

8

A

7

A

Undirected Graph

July, 2007 STATPHYS23 (Genova, Italy) 8

1

A

3

A

2

A

4

A

6

A

5

A

13

ˆ H

67

ˆ H

24

ˆ H

25

ˆ H

346

ˆ H

568

ˆ H

8

A

7

A

A Quantum-Statistical Extension of Probabilistic Inference

( )

∑ ∑ ∑ ∑

∈ = ∈

= − − ≡

B i x i x B a

h a a W a

γ γ γ γ γ

H S Η ˆ log

8 1 r

r r r ( )

∑ ∑

∈

− − − ≡

γ γ γ γ i x i i x a

B h a a W a S H 1 log ˆ

r

r r r

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ≡ 1 1 I

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ≡ 1 1

x

S

{ }

67 , 568 , 346 , 25 , 24 , 13 = B

July, 2007 STATPHYS23 (Genova, Italy) 9

Numerical Results for Quantum Belief Propagation

1

A

3

A

2

A

4

A

6

A

5

A

13

ˆ H

67

ˆ H

24

ˆ H

25

ˆ H

346

ˆ H

568

ˆ H

8

A

7

A Undirected Graph

... 8272 . tr ... 9029 . tr

4 4 1 1

= = = = ρ S S ρ S S

z z z z

... 8379 . tr ... 9032 . tr

4 4 1 1

= = = = ρ S S ρ S S

z z z z

Exact Exact Quantum Quantum Belief Propagation Belief Propagation

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ≡ 1 1

z

S

1 =

x

h

July, 2007 STATPHYS23 (Genova, Italy) 10

Linear Response Theory

} | { Ω x ∈ = i xi

z z z h z z z z z z

h d e e

z

ρ S ρ S S S ρ S S S S

H H 3 3 3 8 1 3 8 3 8

tr ~ tr lim ) tr( : − = 〉 〉〈 〈 − ≡ 〉〉 〈〈

→ −

∫

λ

λ λ

) exp( ~ 1 ~

8 z z

h Z S H ρ + − ≡

1

A

3

A

2

A

4

A

6

A

5

A

8

A

7

A

) exp( 1 H ρ − ≡ Z

July, 2007 STATPHYS23 (Genova, Italy) 11

Numerical Results for Canonical Correlations

1

A

3

A

2

A

4

A

6

A

5

A

13

ˆ H

67

ˆ H

24

ˆ H

25

ˆ H

346

ˆ H

568

ˆ H

8

A

7

A Undirected Graph

... 1727 . : ... 0918 . :

4 8 3 8

= 〉〉 〈〈 = 〉〉 〈〈

z z z z

S S S S

Exact Exact Quantum Quantum Belief Propagation Belief Propagation

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ≡ 1 1

z

S

... 0815 . : ... 0740 . :

4 8 3 8

= 〉〉 〈〈 = 〉〉 〈〈

z z z z

S S S S

1 =

x

h

July, 2007 STATPHYS23 (Genova, Italy) 12

Summary

An Extension to Quantum Belief Propagation Statistical-Mechanical Approaches to Quantum Statistical Inferences Future Problem

It is based on Quantum CVM (Morita, JPSJ1957)