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量子情報処理にむけての クラスター変分法と確率伝搬法の定式化

量子情報処理にむけての クラスター変分法と確率伝搬法の定式化. 田中和之 東北大学大学院情報科学研究科 E-mail: kazu@smapip.is.tohoku.ac.jp. Information Processing and Many Body Problem in Quantum Statistical Mechanics. Quantum Hopfield Model (H. Nishimori) Quantum Annealing (H. Nishimori)

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量子情報処理にむけての クラスター変分法と確率伝搬法の定式化

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  1. 量子情報処理にむけてのクラスター変分法と確率伝搬法の定式化量子情報処理にむけてのクラスター変分法と確率伝搬法の定式化 田中和之 東北大学大学院情報科学研究科 E-mail: kazu@smapip.is.tohoku.ac.jp MSI2005 (Tokyo Institute of Technology)

  2. Information Processing and Many Body Problem in Quantum Statistical Mechanics • Quantum Hopfield Model (H. Nishimori) • Quantum Annealing (H. Nishimori) • Quantum Error Correcting Codes by using Gauge Fields (H. Nishimori) Practical Algorithms for Large-Scale System based on Quantized Probabilistic Model MSI2005 (Tokyo Institute of Technology)

  3. My works of Information Processing by using in Quantum Statistical Mechanics • Quantum Annealing for Image Processing K. Tanaka and T. Horiguchi: Quantum Statistical-Mechanical Iterative Method in Image Restoration, IEICE Transactions (A), J80-A (1997). • Quantized Line Fields in Image Processing K. Tanaka: Image Restorations by using Compound Gauss-Markov Random Field Model with Quantized Line Fields, IEICE Transactions (D-II), J84-D-II (2001). MSI2005 (Tokyo Institute of Technology)

  4. Formulation of Belief Propagation • Link between belief propagation and statistical mechanics. 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 andPractice (MIT Press, 2001). • Generalized belief propagation 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). • Information geometrical interpretation of belief propagation • S. Ikeda, T. Tanaka and S. Amari: Stochastic reasoning, free energy, and information geometry, Neural Computation, 16 (2004). MSI2005 (Tokyo Institute of Technology)

  5. Probabilistic Inference and Quantized Probabilistic Inference • Probabilistic Inference: How should we treat the calculation of the summation over 2N configuration? • Quantized Probabilistic Inference: How should we treat the calculation of the diagonalizaion of 2Nx2N matrix? It is very hard to calculate exactly except some special cases. MSI2005 (Tokyo Institute of Technology)

  6. Cluster Variation Method • Cluster Variation Method for Classical System 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). • Cluster Variation Method for Quantum System T. Morita: Cluster variation method of cooperative phenomena and its generalization II, Quantum Statistics, J. Phys. Soc. Jpn, 12 (1957). MSI2005 (Tokyo Institute of Technology)

  7. Quantum Heisenberg Model 2Nx2N Matrix MSI2005 (Tokyo Institute of Technology)

  8. Gibbs Distribution and Minimization of Free Energy MSI2005 (Tokyo Institute of Technology)

  9. Marginal Density Matrix Consistency Conditions MSI2005 (Tokyo Institute of Technology)

  10. Approximate Free Energy in CVM Consistency Conditions MSI2005 (Tokyo Institute of Technology)

  11. Variational Calculation of Approximate Free Energy MSI2005 (Tokyo Institute of Technology)

  12. Approximate Marginal Density Matrix and Effective Fields Lagrange Multipliers Effective Field from k to i MSI2005 (Tokyo Institute of Technology)

  13. Deterministic Equation of Effective Fields MSI2005 (Tokyo Institute of Technology)

  14. Deterministic Equation of Effective Fields Simple Cubic Lattice High Temperature Expansion + Pade Approximation: 0.8387 (J>0) Ground state is always paramagnetic state in both Bethe and Kikuchi approximations. MSI2005 (Tokyo Institute of Technology)

  15. Simple Cubic Lattice Square Lattice Para Para AF F F 0 0 1 1 Random Heisenberg-Ising Magnets by using Bethe Approximation S. Katsura and K. Shimada: Critical Temperatures of the Random Heisenberg-Ising Magnets, Phys. Stat. Sol. 97 (1980). MSI2005 (Tokyo Institute of Technology)

  16. Extension of Cluster Variation Method • T. Morita: An Approximation Scheme of the Cluster Variation Method for Quantum Lattice Gases, Progress of Theoretical Physics, 92 (1994). Tractable Model MSI2005 (Tokyo Institute of Technology)

  17. Quantum Lattice Gas Model MSI2005 (Tokyo Institute of Technology)

  18. Marginal Density Matrix of Quantum Lattice Gas Model MSI2005 (Tokyo Institute of Technology)

  19. Approximate Free Energy of QCVM in Quantum Lattice Gas Model MSI2005 (Tokyo Institute of Technology)

  20. Quantum Lattice Gas Model Consistency Conditions MSI2005 (Tokyo Institute of Technology)

  21. Quantum Lattice Gas Model Variational Calculation MSI2005 (Tokyo Institute of Technology)

  22. Quantum Lattice Gas Model MSI2005 (Tokyo Institute of Technology)

  23. Quantum Heisenberg Model Holstain-Primakof Transformation Tractable Part Pair Approximation of QCVM T. Morita: A Bose Lattice Gas Equivalent to Heisenberg Model and Its QCVM Study, Journal of the Physical Society of Japan, Vol.64, No.4, pp.1211-1216, 1995. MSI2005 (Tokyo Institute of Technology)

  24. まとめ • 従来型のＣＶＭによる量子系の取り扱い • より一般化されたＣＶＭによる量子系の取り扱い 今後の課題 • 量子推定への応用（ベイズ推定からのアプローチ） • 量子確率伝搬法としてのアルゴリズム • 量子アニーリングへのアプローチ MSI2005 (Tokyo Institute of Technology)

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