physical fluctuomatics 13th quantum-mechanical extensions of probabilistic information processing

Physical Fluctuomatics (Tohoku University) 1

Physical Fluctuomatics13th Quantum-mechanical extensions of probabilistic

information processing

Kazuyuki TanakaGraduate School of Information Sciences, Tohoku University

[email protected]://www.smapip.is.tohoku.ac.jp/~kazu/

Contents1. Introduction2. Quantum System and Density

Matrix3. Transformation between

Density Matrix and Probability Distribution by using Suzuki-Trotter Formula

4. Quantum Belief Propagation5. Summary Physical Fluctuomatics (Tohoku University) 2


Probability Distribution: 2N-tuple summation

1,0 1,0 1,0

211 2

,,,a a a

NN

aaaW

Probability Distribution and Density Matrix

Density Matrix: Diagonalization of 2N× 2N Matrix

VectorEigen 21

ValueEigen

21VectorEigen

21 ,,,),,,(,,, NNN aaaaaaaaa R


Mathematical Framework of Probabilistic Information Processing

32232112321 ,,,, aawaawaaaP

311 3

3223211232122 ,,,,aaa a

aawaawaaaPaP

Such computations are difficult in quantum systems.

BABA expexpexp For any matrices A and B, it is not always valid that


Quantum State of One Node

101 ,010 ,10

1 ,01

0

1

0 0 1

All the possible states in classical Systems are two as follows:

Two vectors in two-dimensional space

1a 0a


Quantum State of One Node

101,010,10

1,01

0

1

0 0

Quantum states are expressed in terms of superpositions of two classical states.

00 1

Quantum states are expressed in terms of any position vectors on unit circle.

Classical States are expressed in terms of two position vectors

10

23

01

21

10

23

01

21

2/3 2/1

2/3

2/1The coefficients can take complex numbers as well as real numbers.

10

01


Probability Distribution

)cosh(2)exp(}Pr{h

hxxX

}1Pr{}1Pr{0}1Pr{}1Pr{0

XXhXXh


Density Matrix

1001

I

11110

110

01001

101

zz

zz

SS

SS

h

h

n

n

n

n

n

nn

n

ee

hn

hn

hh

nh

nh

00

)(!

10

0!

1

00

!1

!1exp

0

0

00

zz SS

10

01zS

)exp(tr )exp(z

z

hh

SSR

)cosh(20

0trexptr h

ee

hh

h

zS

h

h

ee

h 00

)cosh(21R


Quantum State of One Node and Pauli Spin Matrices

0001

00

0010

10

0100

01

1000

11

11001001

I 1100

1001

zS

01100110

xS 0110

00

i

iiyS

101 ,010 ,10

1 ,01

0



T10

010110

UUS x

T0

0exp UUS x

h

h

ee

h

)10(2

1),10(2

1

1111

21U

)cosh(20

0trexptr T h

ee

hh

h

UUS x

)exp(tr )exp(x

x

hh

SSR

T0

0)cosh(2

1 UUR

h

h

ee

h



)10(

21),10(

21

1111

21U

)exp(tr )exp(x

x

hh

SSR

Th

h

ee

hUUR

00

)cosh(21

)cosh(2)10(

21)10(

21

heh

R)cosh(2

)10(2

1)10(2

1h

e h

R

)10(2

1state theofy Probabilit )10(2

1state theofy Probabilit

)10(2

1),10(2

11111

21TU


Quantum State of Two Nodes

,, 12T

2121 aaaaaaa , 2121 aaaaa

Taa

0001

01

0

01

1

01

01

0,0

0010

10

0

10

1

10

01

1,0

101,010,10

1,01

0

0100

01

1

01

0

01

10

0,1

1000

10

1

10

0

10

10

1,1

1 2


Transition Matrix of Two Nodes

0000000000100000

0010

0010

01101,01,0

0000000000000100

0100

0001

10000,10,0

Inner Product of same states provides a diagonal element.

Inner Product of different states provides an off-diagonal element.

1 2


Hamiltonian and Density Matrix

1,1;1,10,1;1,11,0;1,10,0;1,11,1;0,10,1;0,11,0;0,10,0;0,11,1;1,00,1;1,01,0;1,00,0;1,01,1;0,00,1;0,01,0;0,00,0;0,0

;1,0 1,0 1,0 1,0

212121211 2 1 2

uuuuuuuuuuuuuuuu

bbbbaauaaa a b b

H

0 !

1expn

n

nHH

H

Hρ

exptr

exp

Hamiltonian

Density Matrix

1 2


Density Matrix and Probability Distribution

1,1ln00000,1ln00001,0ln00000,0ln

PP

PP

H

1,100000,100001,000000,0

exptrexp

PP

PP

HHR

1,0 1,0

211 2

1,x x

xxP 0, 21 xxPProbability Distribution P(x1,x2)

H is a diagonal matrix and each diagonal element is defined by ln P(x1,x2)

1 2


Computation of Density Matrix

1

3

2

1

0

000000000000

UUH

1

3

2

1

0

exp1000

0exp100

00exp10

000exp1exptr

exp

UU

HHR

Z

Z

Z

Z

3

0

expn

nZ

Statistical quantities of the density matrix can be calculated by diagonalising the Hamiltonian H.

1 2


Probability Distribution and Density MatrixEach state and it corresponding probability

1

3

2

1

0

)exp(0000)exp(0000)exp(0000)exp(

1

exptrexp

UU

HHR

Z

)exp(}Pr{)exp(}Pr{

)exp(}Pr{)exp(}Pr{

333333

222222

111111

000000

uuuuuuuuuuuuuuuu

HHH

H

21, xxP 1,1or 0,1,1,0,0,0 Classical State

Quantum State

3210 ,,, uuuu

U

21


Marginal Probability Distribution and Reduced Density Matrix

RR ii \tr

Marginal Probability Distribution

ix

ii PxP\x

x

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



0,0;0,01,0;0,00,1;0,01,1;0,00,0;1,01,0;1,00,1;1,01,1;1,00,0;0,11,0;0,10,1;0,11,1;0,10,0;1,11,0;1,10,1;1,11,1;1,1

RRRRRRRRRRRRRRRR

R

0,0;0,01,0;1,00,1;0,01,1;1,00,0;0,11,0;1,10,1;0,11,1;1,1

tr 1\1

RRRRRRRR

RR

Partial trace under fixed state at node 1

1 2



1,1;1,10,1;1,11,0;1,10,0;1,11,1;0,10,1;0,11,0;0,10,0;0,11,1;1,00,1;1,01,0;1,00,0;1,01,1;0,00,1;0,01,0;0,00,0;0,0

RRRRRRRRRRRRRRRR

R

1,1;1,11,0;1,00,1;1,10,0;1,01,1;0,11,0;0,00,1;0,100;0,0

tr 2\2

RRRRRRRR

RR

Partial trace under fixed state at node 2

1 2


Quantum Heisenberg Model with Two Nodes

10

01zS

0110xS

00i

iyS

zyx ,, , 21 νν SISISS

zzyyxx JJJ 212121 SSSSSSH

H

HR

exptr

exp

1001

I

1 2


Quantum Heisenberg Model with Two Nodes

10

01zS

0110xS

00i

iyS

1000010000100001

,

1000010000100001

000000

000000

,

000000

000000

0100100000010010

,

0010000110000100

21

21

21

zz SISISS

SISISS

SISISS

zz

yyyy

xxxx

ii

ii

ii

ii

JJJ

JJJ

JJJ

000020020000

212121zzyyxx SSSSSSH

1001

I

1 2


Eigen States of Quantum Heisenberg Model with Two Nodes

100002/12/1002/12/100001

0000300000000

100002/12/1002/12/100001

JJ

JJ

H

State) (Singlet 0,11,02

1

01

10

21 :Eigenstate3 :Eigenvelue

J

State)(Triplet 1,1

1000

,0,11,02

1

0110

21 ,0,0

0001

:sEigenstate :Eigenvelue

J

)exp(1,1Pr)}0,11,0(2

1Pr{}0,0Pr{ J

)3exp()0,11,0(2

1Pr J

H

HR

exptr

exp

1 2


Computation of Density Matrix of Quantum Heisenberg Model with Two Nodes

100002/12/1002/12/100001

0000300000000

100002/12/1002/12/100001

212121

JJ

JJ

JJJ zzyyxx SSSSSSH

J

J

J

J

J

J

J

eJJJJ

e

J

ee

ee

Je

2

2

3

00002cosh2sinh002sinh2cosh0000

2cosh41

100002/12/1002/12/100001

000000000000

100002/12/1002/12/100001

2cosh41

exptrexp

HHR

1 2


Representationon of Ising Model with Two Nodes by Density Matrix

10

01zS zzzz SISISS 21 ,

JJ

JJ

J

000000000000

21zz SSH

1,100001,100001,100001,1

exptrexp

PP

PP

HHR

1001

I

1 121

2121

1 2

expexp

,

x xxJx

xJxxxP

Diagonal Elements correspond to Probability Distribution of Ising Model.

Probability Distribution of Ising Model

Density Matrix

1 2


Transverse Ising Model

10

01zSzzzz SISISS 21 ,

JhhhJhhJh

hhJ

hhJ xxzz

00

00

2121 SSSSH

H

HR

exptr

exp

1001

I

Density Matrix 1 2


Density Matrix of Three Nodes

2312 HHH

23x23 Matrix

H

HR

exptr

exp

1 2 3

1 2 3

1 2 3

=

+



1,1,10,1,11,0,10,0,11,1,00,1,01,0,00,0,0

321 aaa

1,1;1,100,1;1,101,0;1,100,0;1,1001,1;1,100,1;1,101,0;1,100,0;1,1

1,1;0,100,1;0,101,0;0,100,0;0,1001,1;0,100,1;0,101,0;1,100,0;0,1

1,1;1,000,1;1,001,0;1,000,0;1,0001,1;1,000,1;1,001,0;1,000,0;1,0

1,1;0,000,1;0,001,0;0,000,0;0,0001,1;0,000,1;0,001,0;0,000,0;0,0

,;,

12121212

12121212

12121212

12121212

12121212

12121212

12121212

12121212

1,0 1,0 1,0 1,0 1,0 1,0321,21211232112

1 2 3 1 2 333

uuuuuuuu

uuuuuuuu

uuuuuuuu

uuuuuuuu

bbbbbaauaaaa a a b b b

baH

3332112321 0 baaaaHbbb

1 2 3



0,0,01,0,00,1,01,1,00,0,11,0,10,1,11,1,1

321 aaa

1,1;1,10,1;1,11,0;1,10,0;1,100001,1;0,10,1;0,11,0;0,10,0;0,100001,1;1,00,1;1,01,0;1,00,0;1,000001,1;0,00,1;0,01,0;0,00,0;0,00000

00001,1;1,10,1;1,11,0;1,10,0;1,100001,1;0,10,1;0,11,0;0,10,0;0,100001,1;1,00,1;1,01,0;1,00,0;1,000001,1;0,00,1;0,01,0;0,00,0;0,0

,;,

23232323

23232323

23232323

23232323

23232323

23232323

23232323

23232323

1,0 1,0 1,0 1,0 1,0 1,0321,32322332123

1 2 3 1 2 311

uuuuuuuuuuuuuuuu

uuuuuuuuuuuuuuuu

bbbbbaauaaaa a a b b b

baH

1132123321 0 baaaaHbbb

1 2 3

Difficulty of Quantum Systems

2312

2312

exptrexp

HHHHρ

23121223

23122312 expexpexpHHHH

HHHH

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

32Physical Fluctuomatics (Tohoku University)

Suzuki-Trotter Formula

bbcWccWccWcaWa

bbnn

c

cnn

c

cnn

c

cnn

aa

ncccn

n

cccn

n

n

);();();();(lim

1exp1exp

1exp1exp

1exp1exp

1exp1explim

exp

132,,,

211

23121

323122

223121

,,,12312

2312

121

121

HH

HH

HH

HH

HHn: Trotter number



a b cccnn

bbcccaPan

121 ,,,121

2312

2312

,,,,,lim

exptrexp

HHHHρ n: Trotter number

bcccannn

nnnn

n

bcWccWccWcaWbcWccWccWcaWbcccaP

,,,,,112211

112211121

121

);();();();();();();();(,,,,,

Density Matrix ST FormulaΣ

b

a321 ,, ccc

3c

2c

1c




Density Matrix ST FormulaΣ

b

a321 ,, ccc

3c

2c

1c


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



Density Matrix and Reduced Density Matrix

EHH

},{},{

jiji

4

23

１

5

6

7

8

9

}9,6{},8,4{},7,3{},6,1{}5,1{},4,2{},3,2{},2,1{

E

9,8,7,6,5,4,3,2,1V

H

HR

exptr

exp

H{i,j} is a 29×29 matrix.


Density Matrix and Reduced Density Matrix

EHH

},{},{

jiji

HHR

exptr

exp

RR ii \tr RR },{\},{ tr jiji

},{\tr jiii RR


Reducibility Condition


Approximate Expressions of Reduced Density Matrices in Quantum Belief Propagation

iiZ kiki λR exp1

i\jljl

j\ikik λλHR },{},{ exp1

jiij

ji Z

},{\tr jiii RR

ji

ji

i

i j


Message Passing Rule of Quantum Belief Propagation

i\jljl

j\ikik

j\ikikij

λλH

λλ

},{},{

exptrlog ji\iji

iZ

Z

Message Passing Ruleijii ρρ \tr

jiOutput


SummaryProbability Distribution and Density MatrixReduced Density MatrixQuantum Heisenberg ModelSuzuki Trotter FormulaQuantum Belief Propagation


My works of Information Processing by using in Quantum Probabilistic Model and Quantum Belief PropagationK. 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 13th quantum-mechanical extensions of probabilistic information processing

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