1. Physical Fluctuomatics 13th Quantum-mechanical extensions of probabilistic information processing
Kazuyuki Tanaka
Graduate School of Information Sciences, Tohoku University
Physical Fluctuomatics (Tohoku University)

2. Physical Fluctuomatics (Tohoku University)
Contents
Introduction
Quantum System and Density Matrix
Transformation between Density Matrix and Probability Distribution by using Suzuki-Trotter Formula
Quantum Belief Propagation
Summary
Physical Fluctuomatics (Tohoku University)

3. Probability Distribution and Density Matrix
Probability Distribution: 2N-tuple summation
Density Matrix: Diagonalization of 2N× 2N Matrix
Physical Fluctuomatics (Tohoku University)

4. Mathematical Framework of Probabilistic Information Processing
Such computations are difficult in quantum systems.
For any matrices A and B, it is not always valid that
Physical Fluctuomatics (Tohoku University)

5. Physical Fluctuomatics (Tohoku University)
Contents
Introduction
Quantum System and Density Matrix
Transformation between Density Matrix and Probability Distribution by using Suzuki-Trotter Formula
Quantum Belief Propagation
Summary
Physical Fluctuomatics (Tohoku University)

6. Quantum State of One Node
All the possible states in classical Systems are two as follows: 1 1
Two vectors in two-dimensional space
Physical Fluctuomatics (Tohoku University)

7. Quantum State of One Node
Classical States are expressed in terms of two position vectors
Quantum states are expressed in terms of any position vectors on unit circle.
Quantum states are expressed in terms of superpositions of two classical states. 1 1
The coefficients can take complex numbers as well as real numbers.
Physical Fluctuomatics (Tohoku University)

8. Probability Distribution
Physical Fluctuomatics (Tohoku University)

9. Physical Fluctuomatics (Tohoku University)
Density Matrix
Physical Fluctuomatics (Tohoku University)

10. Quantum State of One Node and Pauli Spin Matrices
Physical Fluctuomatics (Tohoku University)

11. Quantum State of One Node and Pauli Spin Matrices
Physical Fluctuomatics (Tohoku University)

12. Quantum State of One Node and Pauli Spin Matrices
Physical Fluctuomatics (Tohoku University)

13. Quantum State of Two Nodes1 2
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14. Transition Matrix of Two Nodes1 2
Inner Product of same states provides a diagonal element.
Inner Product of different states provides an off-diagonal element.
Physical Fluctuomatics (Tohoku University)

15. Hamiltonian and Density Matrix1 2
Hamiltonian
Density Matrix
Physical Fluctuomatics (Tohoku University)

16. Density Matrix and Probability Distribution
Probability Distribution P(x1,x2) 1 2
H is a diagonal matrix and each diagonal element is defined by ln P(x1,x2)
Physical Fluctuomatics (Tohoku University)

17. Computation of Density Matrix1 2
Statistical quantities of the density matrix can be calculated by diagonalising the Hamiltonian H.
Physical Fluctuomatics (Tohoku University)

18. Probability Distribution and Density Matrix1 2
Each state and it corresponding probability
Classical State
Quantum State
Physical Fluctuomatics (Tohoku University)

19. Marginal Probability Distribution and Reduced Density Matrix
Sum of random variables of all the nodes except the node i
Reduced Density Matrix
Partial trace for the freedom of all the nodes except the node i
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20. Reduced Density Matrix1 2
Partial trace under fixed state at node 1
Physical Fluctuomatics (Tohoku University)

21. Reduced Density Matrix1 2
Partial trace under fixed state at node 2
Physical Fluctuomatics (Tohoku University)

22. Quantum Heisenberg Model with Two Nodes1 2
Physical Fluctuomatics (Tohoku University)

23. Quantum Heisenberg Model with Two Nodes1 2
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24. Eigen States of Quantum Heisenberg Model with Two Nodes1 2
Physical Fluctuomatics (Tohoku University)

25. Physical Fluctuomatics (Tohoku University)
Computation of Density Matrix of Quantum Heisenberg Model with Two Nodes 1 2
Physical Fluctuomatics (Tohoku University)

26. Representationon of Ising Model with Two Nodes by Density Matrix1 2
Probability Distribution of Ising Model
Density Matrix
Diagonal Elements correspond to Probability Distribution of Ising Model.
Physical Fluctuomatics (Tohoku University)

27. Transverse Ising Model
Density Matrix 1 2
Physical Fluctuomatics (Tohoku University)

28. Density Matrix of Three Nodes1 2 3 = 1 2 3 + 1 2 3
23x23 Matrix
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29. Density Matrix of Three Nodes1 2 3
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30. Density Matrix of Three Nodes1 2 3
Physical Fluctuomatics (Tohoku University)

31. Physical Fluctuomatics (Tohoku University)
Contents
Introduction
Quantum System and Density Matrix
Transformation between Density Matrix and Probability Distribution by using Suzuki-Trotter Formula
Quantum Belief Propagation
Summary
Physical Fluctuomatics (Tohoku University)

32. Difficulty of Quantum Systems
Addition and Subtraction Formula of Exponential Function is not always valid.
Physical Fluctuomatics (Tohoku University)

33. Suzuki-Trotter Formula
n: Trotter number
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34. Suzuki-Trotter Formula
n: Trotter number
Probability Distribution Σ
ST Formula
Density Matrix
Physical Fluctuomatics (Tohoku University)

35. Suzuki-Trotter Formula
Quantum System on Chain Graph with Three Nodes
Statistical quantities can be computed by using belief propagation of graphical model on 3×n ladder graph
Probability Distribution Σ
ST Formula
Density Matrix
Physical Fluctuomatics (Tohoku University)

36. Physical Fluctuomatics (Tohoku University)
Contents
Introduction
Quantum System and Density Matrix
Transformation between Density Matrix and Probability Distribution by using Suzuki-Trotter Formula
Quantum Belief Propagation
Summary
Physical Fluctuomatics (Tohoku University)

37. Density Matrix and Reduced Density Matrix4 2 3 １ 5 6 7 8 9
H{i,j} is a 29×29 matrix.
Physical Fluctuomatics (Tohoku University)

38. Density Matrix and Reduced Density Matrix
Reducibility Condition
Physical Fluctuomatics (Tohoku University)

39. Physical Fluctuomatics (Tohoku University)
Approximate Expressions of Reduced Density Matrices in Quantum Belief Propagation i i j i j i j
Physical Fluctuomatics (Tohoku University)

40. Message Passing Rule of Quantum Belief Propagationj i
Output
Physical Fluctuomatics (Tohoku University)

41. Physical Fluctuomatics (Tohoku University)
Contents
Introduction
Quantum System and Density Matrix
Transformation between Density Matrix and Probability Distribution by using Suzuki-Trotter Formula
Quantum Belief Propagation
Summary
Physical Fluctuomatics (Tohoku University)

42. Physical Fluctuomatics (Tohoku University)
Summary
Probability Distribution and Density Matrix
Reduced Density Matrix
Quantum Heisenberg Model
Suzuki Trotter Formula
Quantum Belief Propagation
Physical Fluctuomatics (Tohoku University)

43. Physical Fluctuomatics (Tohoku University)
My works of Information Processing by using in Quantum Probabilistic Model and Quantum Belief Propagation
K. Tanaka and T. Horiguchi: Quantum Statistical-Mechanical Iterative Method in Image Restoration, IEICE Transactions (A), vol.J80-A, no.12, pp , December (in Japanese); translated in Electronics and Communications in Japan, Part 3: Fundamental Electronic Science, vol.83, no.3, pp.84-94, March 2000.
K. Tanaka: Image Restorations by using Compound Gauss-Markov Random Field Model with Quantized Line Fields, IEICE Transactions (D-II), vol.J84-D-II, no.4, pp , April (in Japanese); see also Section 5.2 in K. Tanaka, Journal of Physics A: Mathematical and General, vol.35, no.37 , pp.R81-R150, September 2002.
K. Tanaka: Mathematical Structures of Loopy Belief Propagation and Cluster Variation Method, Journal of Physics: Conference Series, vol.143, article no , pp.1-18, January 2009
Physical Fluctuomatics (Tohoku University)