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

Contents 1. Introduction 2. Quantum System and Density Matrix 3. Transformation between Density Matrix and Probability Distribution by using Suzuki-Trotter Formula 4. Quantum Belief Propagation 5. Summary Physical Fluctuomatics (Tohoku University)2

3 Probability Distribution: 2 N - tuple summation Probability Distribution and Density Matrix Density Matrix: Diagonalization of 2 N × 2 N 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

Contents 1. Introduction 2. Quantum System and Density Matrix 3. Transformation between Density Matrix and Probability Distribution by using Suzuki-Trotter Formula 4. Quantum Belief Propagation 5. Summary Physical Fluctuomatics (Tohoku University)5

6 Quantum State of One Node All the possible states in classical Systems are two as follows: Two vectors in two-dimensional space

Physical Fluctuomatics (Tohoku University)7 Quantum State of One Node 10 0 Quantum states are expressed in terms of superpositions of two classical states Quantum states are expressed in terms of any position vectors on unit circle. Classical States are expressed in terms of two position vectors The coefficients can take complex numbers as well as real numbers.

Physical Fluctuomatics (Tohoku University)8 Probability Distribution

Physical Fluctuomatics (Tohoku University)9 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 Nodes 12

Physical Fluctuomatics (Tohoku University)14 Transition Matrix of Two Nodes Inner Product of same states provides a diagonal element. Inner Product of different states provides an off-diagonal element. 12

Physical Fluctuomatics (Tohoku University)15 Hamiltonian and Density Matrix Hamiltonian Density Matrix 12

Physical Fluctuomatics (Tohoku University)16 Density Matrix and Probability Distribution Probability Distribution P(x 1,x 2 ) H is a diagonal matrix and each diagonal element is defined by ln P(x 1,x 2 ) 12

Physical Fluctuomatics (Tohoku University)17 Computation of Density Matrix Statistical quantities of the density matrix can be calculated by diagonalising the Hamiltonian H. 12

Physical Fluctuomatics (Tohoku University)18 Probability Distribution and Density Matrix Each state and it corresponding probability Classical State Quantum State 21

Physical Fluctuomatics (Tohoku University)19 Marginal Probability Distribution and Reduced Density Matrix Marginal Probability Distribution Reduced Density Matrix Sum of random variables of all the nodes except the node i Partial trace for the freedom of all the nodes except the node i

Physical Fluctuomatics (Tohoku University)20 Reduced Density Matrix Partial trace under fixed state at node 1 1 2

Physical Fluctuomatics (Tohoku University)21 Reduced Density Matrix Partial trace under fixed state at node 2 12

Physical Fluctuomatics (Tohoku University)22 Quantum Heisenberg Model with Two Nodes 12

Physical Fluctuomatics (Tohoku University)23 Quantum Heisenberg Model with Two Nodes 12

Physical Fluctuomatics (Tohoku University)24 Eigen States of Quantum Heisenberg Model with Two Nodes 12

Physical Fluctuomatics (Tohoku University)25 Computation of Density Matrix of Quantum Heisenberg Model with Two Nodes 12

Physical Fluctuomatics (Tohoku University)26 Representationon of Ising Model with Two Nodes by Density Matrix Diagonal Elements correspond to Probability Distribution of Ising Model. Probability Distribution of Ising Model Density Matrix 12

Physical Fluctuomatics (Tohoku University)27 Transverse Ising Model Density Matrix 12

Physical Fluctuomatics (Tohoku University)28 Density Matrix of Three Nodes 2 3 x2 3 Matrix = +

Physical Fluctuomatics (Tohoku University)29 Density Matrix of Three Nodes 12 3

Physical Fluctuomatics (Tohoku University)30 Density Matrix of Three Nodes 1 23

Contents 1. Introduction 2. Quantum System and Density Matrix 3. Transformation between Density Matrix and Probability Distribution by using Suzuki-Trotter Formula 4. Quantum Belief Propagation 5. Summary Physical Fluctuomatics (Tohoku University)31

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

Suzuki-Trotter Formula n: Trotter number 33Physical Fluctuomatics (Tohoku University)

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

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

Contents 1. Introduction 2. Quantum System and Density Matrix 3. Transformation between Density Matrix and Probability Distribution by using Suzuki-Trotter Formula 4. Quantum Belief Propagation 5. Summary Physical Fluctuomatics (Tohoku University)36

Physical Fluctuomatics (Tohoku University)37 Density Matrix and Reduced Density Matrix １ H {i,j} is a 2 9 ×2 9 matrix.

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

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

Physical Fluctuomatics (Tohoku University)40 Message Passing Rule of Quantum Belief Propagation Message Passing Rule j i Output

Contents 1. Introduction 2. Quantum System and Density Matrix 3. Transformation between Density Matrix and Probability Distribution by using Suzuki-Trotter Formula 4. Quantum Belief Propagation 5. Summary Physical Fluctuomatics (Tohoku University)41

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

Physical Fluctuomatics (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 , December 1997 (in Japanese); translated in Electronics and Communications in Japan, Part 3: Fundamental Electronic Science, vol.83, no.3, pp.84-94, March 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 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 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