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

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

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### Physical FluctuomaticsApplied Stochastic Process 12th Quantum-mechanical extensions of probabilistic information processing

### Density Matrix

### Quantum State of One Node and Pauli Spin Matrices

### Quantum State of One Node and Pauli Spin Matrices

ContentsContents

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)

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)

Probability Distribution and Density Matrix

- Probability Distribution: 2N-tuple summation

Density Matrix: Diagonalization of 2N× 2N Matrix

Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

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)

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)

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)

0

0

0

1

Quantum State of One NodeClassical 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)

Probability Distribution

Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

Quantum State of One Node and Pauli Spin Matrices

Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

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)

Hamiltonian and Density Matrix

1

2

Hamiltonian

Density Matrix

Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

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)

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)

Probability Distribution and Density Matrix

1

2

Classical State

Each state and it corresponding probability

Quantum State

Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

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)

2

Reduced Density MatrixPartial trace under fixed state at node 1

Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

2

Reduced Density MatrixPartial trace under fixed state at node 2

Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

Quantum Heisenberg Model with Two Nodes

1

2

Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

Quantum Heisenberg Model with Two Nodes

1

2

Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

Eigen States of Quantum Heisenberg Model with Two Nodes

1

2

Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

Computation of Density Matrix of Quantum Heisenberg Model with Two Nodes

1

2

Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

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)

Transverse Ising Model

Density Matrix

1

2

Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

Density Matrix of Three Nodes

1

2

3

=

1

2

3

+

1

2

3

23x23 Matrix

Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

Density Matrix of Three Nodes

1

2

3

Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

Density Matrix of Three Nodes

1

2

3

Physical Fluctuomatics / Applied Stochastic Process (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 / Applied Stochastic Process (Tohoku University)

Difficulty of Quantum Systems

Addition and Subtraction Formula of Exponential Function is not always valid.

Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

Suzuki-Trotter Formula

n: Trotter number

Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

Suzuki-Trotter Formula

n: Trotter number

Probability Distribution

ST Formula

Density Matrix

Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

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)

- 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)

5

4

１

2

3

6

9

7

Density Matrix and Reduced Density MatrixH{i,j} is a 29×29 matrix.

Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

Density Matrix and Reduced Density Matrix

Reduced Density Matrix

Reducibility Condition

Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

Approximate Expressions of Reduced Density Matrices in Quantum Belief Propagation

i

i

j

i

j

i

j

Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

Message Passing Rule of Quantum Belief Propagation

Message Passing Rule

j

i

Output

Physical Fluctuomatics / Applied Stochastic Process (Tohoku University)

- 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)

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)

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)

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