Skip to main content
SpringerLink
Log in

Finding Exponential Product Formulas of Higher Orders

  • Chapter
  • First Online:
Quantum Annealing and Other Optimization Methods

Part of the book series: Lecture Notes in Physics ((LNP,volume 679))

Abstract

In the present article, we review the progress in the last two decades of the work on the Suzuki-Trotter decomposition, or the exponential product formula. The simplest Suzuki-Trotter decomposition, or the well-known Trotter decomposition [1–xs4] is given by

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M. Suzuki: Commun. Math. Phys. 51, 183 (1976)

    Article  MATH  ADS  Google Scholar 

  2. M. Suzuki: Commun. Math. Phys. 57, 193 (1977)

    Article  MATH  ADS  Google Scholar 

  3. M. Suzuki: Prog. Theor. Phys. 56, 1454 (1976)

    Article  ADS  MATH  Google Scholar 

  4. M. Suzuki: J. Math. Phys. 26, 601 (1985)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  5. M. Suzuki: Phys. Lett. A 146, 319 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  6. M. Suzuki: J. Math. Phys. 32, 400 (1991)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  7. M. Suzuki: Phys. Lett. A 165, 387 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  8. M. Suzuki: J. Phys. Soc. Jpn. 61, 3015 (1992)

    Article  ADS  Google Scholar 

  9. M. Suzuki: Physica A 191, 501 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  10. M. Suzuki: Proc. Jpn. Acad. 69 B, 161 (1993)

    Article  ADS  Google Scholar 

  11. K. Umeno, M. Suzuki: Phys. Lett. A 181, 387 (1993)

    Article  ADS  Google Scholar 

  12. M. Suzuki, K. Umeno: Higher-order decomposition theory of exponential operators and its applications to QMC and nonlinear dynamics. In: Computer Simulation Studies in Condensed-Matter Physics VI, ed by D.P. Landau, K.K. Mon, H.-B. Schüttler (Springer, Berlin Heidelberg, 1993) pp 74-86

    Google Scholar 

  13. M. Suzuki: Physica A 194, 432 (1993)

    Article  ADS  Google Scholar 

  14. M. Suzuki: Physica A 205, 65 (1994)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  15. M. Suzuki: Commun. Math. Phys. 163, 491 (1994)

    Article  MATH  ADS  Google Scholar 

  16. H. Kobayashi, N. Hatano, M. Suzuki: Physica A 211, 234 (1994)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. M. Suzuki: Phys. Lett. A 180, 232 (1993)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  18. M. Suzuki: Convergence of exponential product formula and its applications to Hamiltonian systems. In: Dynamical Systems and Chaos, vol 2, ed by Y. Aizawa, S. Saito, K. Shiraiwa (World Scientific, Singapore, 1994) pp 450-453

    Google Scholar 

  19. M. Suzuki: Exponential product formula and Lie algebra. In: Group Theoretical Methods in Physics, ed by A. Arima, T. Eguchi, N. Nakanishi (World Scientific, Singapore, 1995) pp 459-464

    Google Scholar 

  20. M. Suzuki: Phys. Lett. A 201, 425 (1995)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  21. M. Suzuki: New scheme of hybrid exponential product formulas with applications to quantum Monte Carlo simulations. In: Computer Simulation Studies in Condensed-Matter Physics VIII, ed by D.P. Landau, K.K. Mon, H.-B. Schüttler (Springer, Berlin Heidelberg New York, 1995) pp 169-174

    Google Scholar 

  22. M. Suzuki: General theory of exponential product formulas. In: Computational Physics as a New Frontier in Condensed Matter Research, ed by H. Takayama, M. Tsukada, H. Shiba, F. Yonezawa, M. Imada, Y. Okabe (Physical Society of Japan, Tokyo, 1995) pp 51-56

    Google Scholar 

  23. M. Suzuki: Systematics and numerics in many-body systems. In: Recent Progress in Many-Body Theories, vol 4, ed by E. Schachinger, H. Mitter, H. Sormann (Plenum Press, New York, 1995) pp 65-70

    Google Scholar 

  24. M. Suzuki: General theory of exponential product formulas and its applications to quantum fluctuation. In: Coherent Approaches to Fluctuations, ed by M. Suzuki, N. Kawashima (World Scientific, Singapore, 1996) pp 95-100

    Google Scholar 

  25. Z. Tsuboi, M. Suzuki: Int. J. Mod. Phys. B 9, 3241 (1995)

    Article  MathSciNet  ADS  Google Scholar 

  26. M. Suzuki: Rev. Math. Phys. 8, 487 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  27. M. Suzuki: Int. J. Mod. Phys. B 10, 1637 (1996)

    Article  ADS  Google Scholar 

  28. M. Suzuki: Int. J. Mod. Phys. C 7, 355 (1996)

    Article  ADS  Google Scholar 

  29. M. Suzuki: Commun. Math. Phys. 183, 339 (1997)

    Article  MATH  ADS  Google Scholar 

  30. M. Suzuki: J. Math. Phys. 38, 1183 (1997)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  31. M. Suzuki: Phys. Lett. A 224, 337 (1997)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  32. M. Suzuki: Prog. Theor. Phys. 100, 475 (1998)

    Article  ADS  Google Scholar 

  33. M. Suzuki: Int. J. Mod. Phys. C 10, 1385 (1999)

    Article  MATH  ADS  Google Scholar 

  34. M. Suzuki: Rev. Math. Phys. 11, 243 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  35. M. Suzuki: Comp. Phys. Commun. 127, 32 (2000)

    Article  ADS  MATH  Google Scholar 

  36. M. Suzuki: J. Stat. Phys. 110, 945 (2003)

    Article  MATH  Google Scholar 

  37. M. Suzuki: Physica A 321, 334 (2003)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  38. N. Hatano, M. Suzuki: Prog. Theor. Phys. 85, 481 (1991)

    Article  ADS  Google Scholar 

  39. N. Kawashima, K. Harada: J. Phys. Soc. Jpn. 73, 1379 (2004)

    Article  MATH  ADS  Google Scholar 

  40. T. Sato: Simulated annealing using quantum fluctuation. Master Thesis, University of Tokyo, Tokyo (1995); T. Sato, N. Hatano, M. Suzuki, H. Takayama: unpublished

    Google Scholar 

  41. T. Kadowaki, H. Nishimori: Phys. Rev. E 58, 5355 (1998)

    Article  ADS  Google Scholar 

  42. B.K. Chakrabarti: Transverse Ising Model, Glass and Quantum Annealing. In: Lect. Notes Phys. 679 (2005)

    Google Scholar 

  43. R.D. Ruth: IEEE Trans. Nucl. Sci. 30, 2669 (1983)

    Article  ADS  Google Scholar 

  44. R.I. McLachlan: BIT 35, 258 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  45. H. De Raedt, A. Lagendijk: Phys. Rep. 127, 233 (1985)

    Article  ADS  MathSciNet  Google Scholar 

  46. M. Suzuki (ed): Quantum Monte Carlo Methods in Equilibrium and Nonequilibrium Systems (Springer, Berlin, 1987)

    Google Scholar 

  47. M. Suzuki (ed): Quantum Monte Carlo Methods in Condensed-Matter Physics (World Scientific, Singapore, 1993)

    Google Scholar 

  48. B.K. Chakrabarti, A. Dutta, P. Sen: Quantum Ising Phases and Transitions in Transverse Ising Models. In: Lect. Notes Phys. m41 (1996)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Industrial Science, University of Tokyo, 4–6–1 Meguro, Tokyo, Komaba, 162-8601, Japan

    Naomichi Hatano

  2. Department of Applied Physics, Tokyo University of Science, 1–3 Kagurazaka, Tokyo, Shinjuku, 162-8601, Japan

    Masuo Suzuki

Authors
  1. Naomichi Hatano
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Masuo Suzuki
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Arnab Das Bikas K. Chakrabarti

Rights and permissions

Reprints and permissions

About this chapter

Cite this chapter

Hatano, N., Suzuki, M. Finding Exponential Product Formulas of Higher Orders. In: Das, A., K. Chakrabarti, B. (eds) Quantum Annealing and Other Optimization Methods. Lecture Notes in Physics, vol 679. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11526216_2

Download citation

  • DOI: https://doi.org/10.1007/11526216_2

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-27987-7

  • Online ISBN: 978-3-540-31515-5

  • eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)

Publish with us

Policies and ethics

Search

Navigation