Physical Fluctuomatics Applied Stochastic Process 12th Quantum-mechanical extensions of probabilistic information proce

1. Physical FluctuomaticsApplied Stochastic Process 12th Quantum-mechanical extensions of probabilistic information processing Kazuyuki Tanaka Graduate School of Information Sciences, Tohoku University kazu@smapip.is.tohoku.ac.jp http://www.smapip.is.tohoku.ac.jp/~kazu/ Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

2. 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 / Applied Stochastic Process (Tohoku University)

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

5. 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 / Applied Stochastic Process (Tohoku University)

6. Quantum State of One Node All the possible states in classical Systems are two as follows: 1 0 0 1 Two vectors in two-dimensional space Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

7. 1 0 0 0 1 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. 0 The coefficients can take complex numbers as well as real numbers. Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

8. Probability Distribution Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

9. Density Matrix Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

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

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

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

13. Quantum State of Two Nodes 1 2 Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

14. Transition Matrix of Two Nodes 1 2 Inner Product of same states provides a diagonal element. Inner Product of different states provides an off-diagonal element. Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

15. Hamiltonian and Density Matrix 1 2 Hamiltonian Density Matrix Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

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

17. Computation of Density Matrix 1 2 Statistical quantities of the density matrix can be calculated by diagonalising the Hamiltonian H. Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

18. Probability Distribution and Density Matrix 1 2 Classical State Each state and it corresponding probability Quantum State Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

19. Marginal Probability Distribution and Reduced Density Matrix Marginal Probability Distribution 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 Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

20. 1 2 Reduced Density Matrix Partial trace under fixed state at node 1 Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

21. 1 2 Reduced Density Matrix Partial trace under fixed state at node 2 Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

22. Quantum Heisenberg Model with Two Nodes 1 2 Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

23. Quantum Heisenberg Model with Two Nodes 1 2 Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

24. Eigen States of Quantum Heisenberg Model with Two Nodes 1 2 Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

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

26. Representationon of Ising Model with Two Nodes by Density Matrix 1 2 Probability Distribution of Ising Model Diagonal Elements correspond to Probability Distribution of Ising Model. Density Matrix Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

27. Transverse Ising Model Density Matrix 1 2 Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

28. Density Matrix of Three Nodes 1 2 3 = 1 2 3 + 1 2 3 23x23 Matrix Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

29. Density Matrix of Three Nodes 1 2 3 Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

30. Density Matrix of Three Nodes 1 2 3 Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

31. 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 / Applied Stochastic Process (Tohoku University)

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

33. Suzuki-Trotter Formula n: Trotter number Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

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

35. Σ 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 ST Formula Density Matrix Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

36. 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 / Applied Stochastic Process (Tohoku University)

37. 8 5 4 １ 2 3 6 9 7 Density Matrix and Reduced Density Matrix H{i,j} is a 29×29 matrix. Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

38. Density Matrix and Reduced Density Matrix Reduced Density Matrix Reducibility Condition Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

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

40. Message Passing Rule of Quantum Belief Propagation Message Passing Rule j i Output Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

41. 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 / Applied Stochastic Process (Tohoku University)

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

43. 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.2117-2126, December 1997 (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.737-743, April 2001 (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.012023, pp.1-18, January 2009 Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)