?????????????? Physical Fluctuomatics ?13? ??????????????? 13th Quantum-mechanical extensions of probabilistic information processing - PowerPoint PPT Presentation

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?????????????? Physical Fluctuomatics ?13? ??????????????? 13th Quantum-mechanical extensions of probabilistic information processing

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Title: ?????????????? Physical Fluctuomatics ?13? ??????????????? 13th Quantum-mechanical extensions of probabilistic information processing


1
??????????????Physical Fluctuomatics?13?
??????????????? 13th Quantum-mechanical
extensions of probabilistic information processing
  • ???? ?????????? ????????
  • ?? ??(Kazuyuki Tanaka)
  • kazu_at_smapip.is.tohoku.ac.jp
  • http//www.smapip.is.tohoku.ac.jp/kazu/

????Webpage http//www.smapip.is.tohoku.ac.jp/ka
zu/PhysicalFluctuomatics/2007/
2
Contents
  1. ????
  2. ??????
  3. Suzuki-Trotter ????????????
  4. ????????????????????????
  5. ??????????????????
  6. ???

3
??????????????????
  • ????????????? (H. Nishimori)
  • ???????? (H. Nishimori)
  • ???????????????? (H. Nishimori)

???????????????????
4
My works of Information Processing by using in
Quantum Statistical Mechanics
  • ???????????????????
  • K. Tanaka and T. Horiguchi Quantum
    Statistical-Mechanical Iterative Method in Image
    Restoration, IEICE Transactions (A), J80-A
    (1997).
  • ?????????????????
  • K. Tanaka Image Restorations by using Compound
    Gauss-Markov Random Field Model with Quantized
    Line Fields, IEICE Transactions (D-II), J84-D-II
    (2001).

5
???????????
  • ??????????????????
  • Y. Kabashima and D. Saad, Belief propagation vs.
    TAP for decoding corrupted messages, Europhys.
    Lett. 44 (1998).
  • M. Opper and D. Saad (eds), Advanced Mean Field
    Methods ---Theory and Practice (MIT Press, 2001).
  • ??????????????????????????
  • J. S. Yedidia, W. T. Freeman and Y. Weiss
    Constructing free-energy approximations and
    generalized belief propagation algorithms, IEEE
    Transactions on Information Theory, 51 (2005).
  • ?????????????
  • S. Ikeda, T. Tanaka and S. Amari Stochastic
    reasoning, free energy, and information geometry,
    Neural Computation, 16 (2004).

6
???????????
  • ???? 2N ????????
  • ????? 2N ? 2N ????????

???????????????????????????
7
??????????
???????????
  • ???????????????!!

?? A, B ????
???????????
8
??????
  • 1??????????????????????
  • ????????????????

9
Contents
  1. ????
  2. ??????
  3. Suzuki-Trotter ????????????
  4. ????????????????????????
  5. ??????????????????
  6. ???

10
1????????
????????????2??????
1
0
0
1
2?????2????????
11
1????????
????? 2???? ?????
????? 2???? ?2?
?????2????????????? ????????????????
0
?????????????
12
1????????
???
13
1???????????????
14
1???????????????
15
1???????????????
16
2????????
???????????4?????4?
?????4?????????
??? N ????????? 2N ?????????
17
2?????????????
???????????????????1??1???
?????????????????????1??1???
18
?????????
???????
????
19
?????????
???? P(a1,a2)
H ????????,???????????????????,???????????????????
????.
20
???????
?????????????????????????????.
21
?????????
  • ??????????????

??? ???
???? ??? ???
?????????? ???????????
22
???????????
????
i ??????????????????????
??????
i ???????????????????????
23
??????(Reduced Density Matrix)
???1?????????????????
24
??????(Reduced Density Matrix)
???2?????????????????
25
2 ?????? Heisenberg ???
26
2?????? Heisenberg ???
27
2?????? Heisenberg ????????
28
2?????? Heisenberg ???????????
29
2 ???? Ising ????????????????
Ising ????????
????
????????? ?Ising???? ???????? ?????.
30
2 ??????? Ising ???(Transverse Ising
Model)???????????
????
31
3???????????
23x23 Matrix
32
3???????????
33
3???????????
34
Contents
  1. ????
  2. ??????
  3. Suzuki-Trotter ????????????
  4. ????????????????????????
  5. ??????????????????
  6. ???

35
??????????
?????????????????
36
???????
????????????????
37
Suzuki-Trotter??
n Trotter ?
3xn ??????????????????????
????????n????????????????????????????.
Suzuki-Trotter ??
????
38
Suzuki-Trotter??
n Trotter ?
39
Suzuki-Trotter??
n Trotter ?
40
Suzuki-Trotter??
n Trotter ?
????
ST ??
????
41
Suzuki-Trotter??
3???????1???????????
3xn ???????????????????????????????????????????.
????
ST ??
????
42
Contents
  1. ????
  2. ??????
  3. Suzuki-Trotter ????????????
  4. ????????????????????????
  5. ??????????????????
  6. ???

43
????????(Cluster Variation Method)
  • ????????????
  • R. Kikuchi A theory of cooperative phenomena,
    Phys. Rev., 81 (1951).
  • T. Morita Cluster variation method of
    cooperative phenomena and its generalization I,
    J. Phys. Soc. Jpn, 12 (1957).
  • ????????????
  • T. Morita Cluster variation method of
    cooperative phenomena and its generalization II,
    Quantum Statistics, J. Phys. Soc. Jpn, 12 (1957).
  • T. Morita An Approximation Scheme of the
    Cluster Variation Method for Quantum Lattice
    Gases, Progress of Theoretical Physics, 92 (1994).

44
?????????????????
45
???????????
i ????
ij ? i ? j ??????
B ????????? ??????????
?????? (Reduced Density Matrix)
Reducibility Condition
Normalization Condition
46
?????????????????????
Consistency Conditions
47
????????????
48
????????????
k ?? i ?????
49
????????????????????
????????
j
i
Output
50
Contents
  1. ????
  2. ??????
  3. Suzuki-Trotter ????????????
  4. ????????????????????????
  5. ??????????????????
  6. ???

51
?????????????
  1. ??????????????????????????????.
  2. ????????????????????????????.

52
????????????????????
?? ???
53
?????????
54
??????????????????
55
??????????????????
56
??????????????????
57
????????????????
58
???????????????????????
Exact
Quantum CVM
59
??????
(???????????????????????????????????.)
60
???????????????????????
Exact
Quantum CVM
61
Contents
  1. ????
  2. ??????
  3. Suzuki-Trotter ????????????
  4. ????????????????????????
  5. ??????????????????
  6. ???

62
???
  • ????CVM??????????
  • ???????????????????????????????????

?????
  • ????????????????????
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