Physical Fluctuomatics 13th Quantum-mechanical extensions of probabilistic information processing
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Transcript of 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
Physical Fluctuomatics (Tohoku University) 3
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
Physical Fluctuomatics (Tohoku University) 4
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
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) 5
Physical Fluctuomatics (Tohoku University) 6
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
Physical Fluctuomatics (Tohoku University) 7
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
Physical Fluctuomatics (Tohoku University) 8
Probability Distribution
)cosh(2)exp(}Pr{h
hxxX
}1Pr{}1Pr{0}1Pr{}1Pr{0
XXhXXh
Physical Fluctuomatics (Tohoku University) 9
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
Physical Fluctuomatics (Tohoku University) 10
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
Physical Fluctuomatics (Tohoku University) 11
Quantum State of One Node and Pauli Spin Matrices
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
Physical Fluctuomatics (Tohoku University) 12
Quantum State of One Node and Pauli Spin Matrices
)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
Physical Fluctuomatics (Tohoku University) 13
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
Physical Fluctuomatics (Tohoku University) 14
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
Physical Fluctuomatics (Tohoku University) 15
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
Physical Fluctuomatics (Tohoku University) 16
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
Physical Fluctuomatics (Tohoku University) 17
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
Physical Fluctuomatics (Tohoku University) 18
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
Physical Fluctuomatics (Tohoku University) 19
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
Physical Fluctuomatics (Tohoku University) 20
Reduced Density Matrix
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
Physical Fluctuomatics (Tohoku University) 21
Reduced 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
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
Physical Fluctuomatics (Tohoku University) 22
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
Physical Fluctuomatics (Tohoku University) 23
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
Physical Fluctuomatics (Tohoku University) 24
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
Physical Fluctuomatics (Tohoku University) 25
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
Physical Fluctuomatics (Tohoku University) 26
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
Physical Fluctuomatics (Tohoku University) 27
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
Physical Fluctuomatics (Tohoku University) 28
Density Matrix of Three Nodes
2312 HHH
23x23 Matrix
H
HR
exptr
exp
1 2 3
1 2 3
1 2 3
=
+
Physical Fluctuomatics (Tohoku University) 29
Density Matrix of Three Nodes
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
Physical Fluctuomatics (Tohoku University) 30
Density Matrix of Three Nodes
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
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) 31
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
33Physical Fluctuomatics (Tohoku University)
Suzuki-Trotter Formula
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
Probability Distribution
34Physical Fluctuomatics (Tohoku University)
Suzuki-Trotter Formula
Density Matrix ST FormulaΣ
b
a321 ,, ccc
3c
2c
1c
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)
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) 36
Physical Fluctuomatics (Tohoku University) 37
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.
Physical Fluctuomatics (Tohoku University) 38
Density Matrix and Reduced Density Matrix
EHH
},{},{
jiji
HHR
exptr
exp
RR ii \tr RR },{\},{ tr jiji
},{\tr jiii RR
Reduced Density Matrix
Reducibility Condition
Physical Fluctuomatics (Tohoku University) 39
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
Physical Fluctuomatics (Tohoku University) 40
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
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) 41
Physical Fluctuomatics (Tohoku University) 42
SummaryProbability Distribution and Density MatrixReduced Density MatrixQuantum Heisenberg ModelSuzuki Trotter FormulaQuantum Belief Propagation
Physical Fluctuomatics (Tohoku University) 43
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