Quantum mechanical hamiltonian models of turing machines

Abstract

Quantum mechanical Hamiltonian models, which represent an aribtrary but finite number of steps of any Turing machine computation, are constructed here on a finite lattice of spin-1/2 systems. Different regions of the lattice correspond to different components of the Turing machine (plus recording system). Successive states of any machine computation are represented in the model by spin configuration states. Both time-independent and time-dependent Hamiltonian models are constructed here. The time-independent models do not dissipate energy or degrade the system state as they evolve. They operate close to the quantum limit in that the total system energy uncertainty/computation speed is close to the limit given by the time-energy uncertainty relation. However, the model evolution is time global and the Hamiltonian is more complex. The time-dependent models do not degrade the system state. Also they are time local and the Hamiltonian is less complex.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Jacob D. Bekenstein,Phys. Rev. Lett. 46:623 (1981).

    Google Scholar 

  2. 2.

    Charles H. Bennett,IBM J. Res. Dev. 17:525 (1973).

    Google Scholar 

  3. 3.

    Hans J. Bremerman, “Part I. Limitations on Data Processing Arising From Quantum Theory,” inSelf-Organizing Systems, M. C. Yovits, G. T. Jacobi, and G. D. Goldstein, eds. (Spartan Books, Washington, D.C., 1967).

    Google Scholar 

  4. 4.

    Hans J. Bremerman, “Energy and Entropy Costs in Information Transfer and Computing,” Preprint to appear in Proceedings of Conference on Physics of Computation, MIT, May 6–8, 1981.

  5. 5.

    David Deutsch,Phys. Rev. Lett. 48:286 (1982).

    Google Scholar 

  6. 6.

    Robert W. Keyes,Proc. IEEE 69:267 (1981); Rolf Landauer, ‘Fundamental Physical Limitations of the Computational Process,” Preprint, 1981;Ber. Bunsenges. Phys. Chem. 80:1048(1976).

    Google Scholar 

  7. 7.

    Rolf Landauer,IBM J. Res. Dev. 5:183 (1961); Rolf Landauer and James W. F. Woo,J. Appl. Phys. 42:2301 (1971); Robert W. Keyes and Rolf Landauer,IBM J. Res. Dev. 14:152(1970).

    Google Scholar 

  8. 8.

    K. K. Likharev, “Classical and Quantum Limitations on Energy Consumption in Computation,” to appear inProceedings of Conference on Physics of Computation, MIT, May 6–8, 1981.

  9. 9.

    L. B. Levitin, ‘Physical Limitations of Rate, Depth, and Minimum Energy in Information Processing,” Preprint, Bielefeld University.

  10. 10.

    D. Mundici,Nuovo Cimento 61B:297 (1981).

    Google Scholar 

  11. 11.

    Rolf Landauer, Preprint, to appear in Proceedings of Conference on Physics of Computation, MIT, May 6–8, 1981Int. J. Theor. Phys. 21:283 (1982).

    Google Scholar 

  12. 12.

    Edward Fredkin and Tommaso Toffoli, “Conservative Logic,” Tech. Memo., MIT/LCS/TM-197, April 29, 1981.

  13. 13.

    Paul Benioff,J. Stat. Phys. 22:563 (1980).

    Google Scholar 

  14. 14.

    Paul Benioff,J. Math. Phys. 22:495 (1981).

    Google Scholar 

  15. 15.

    Paul Benioff, “Quantum Mechanical Hamiltonian Models of Discrete Processes that Erase Their Own Histories: Application to Turing Machines,” in Proceedings of Conference on Physics of Computation MIT, May 6–8, 1981.

  16. 16.

    Martin Davis,Computability and Unsolvability (McGraw Hill, New York, 1958).

    Google Scholar 

  17. 17.

    Hartley Rogers, Jr.,Theory of Recursive Functions and Effective Computability (McGraw Hill, New York, 1965), pp. 64,65.

    Google Scholar 

  18. 18.

    A. Messiah,Quantum Mechanics, Vol. I (John Wiley and Sons, New York, 1958), Chap. IV, Sec. 10.

    Google Scholar 

  19. 19.

    Rolf Landauer, Informal Notes, 1981.

Download references

Author information

Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Benioff, P. Quantum mechanical hamiltonian models of turing machines. J Stat Phys 29, 515–546 (1982). https://doi.org/10.1007/BF01342185

Download citation

Key words

  • Schrödinger equation description of Turing machines
  • nondissipative models of computers
  • quantum spin lattices