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, log in via an institution to check access.
Similar content being viewed by others
References
Jacob D. Bekenstein,Phys. Rev. Lett. 46:623 (1981).
Charles H. Bennett,IBM J. Res. Dev. 17:525 (1973).
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).
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.
David Deutsch,Phys. Rev. Lett. 48:286 (1982).
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).
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).
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.
L. B. Levitin, ‘Physical Limitations of Rate, Depth, and Minimum Energy in Information Processing,” Preprint, Bielefeld University.
D. Mundici,Nuovo Cimento 61B:297 (1981).
Rolf Landauer, Preprint, to appear in Proceedings of Conference on Physics of Computation, MIT, May 6–8, 1981Int. J. Theor. Phys. 21:283 (1982).
Edward Fredkin and Tommaso Toffoli, “Conservative Logic,” Tech. Memo., MIT/LCS/TM-197, April 29, 1981.
Paul Benioff,J. Stat. Phys. 22:563 (1980).
Paul Benioff,J. Math. Phys. 22:495 (1981).
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.
Martin Davis,Computability and Unsolvability (McGraw Hill, New York, 1958).
Hartley Rogers, Jr.,Theory of Recursive Functions and Effective Computability (McGraw Hill, New York, 1965), pp. 64,65.
A. Messiah,Quantum Mechanics, Vol. I (John Wiley and Sons, New York, 1958), Chap. IV, Sec. 10.
Rolf Landauer, Informal Notes, 1981.
Author information
Authors and Affiliations
Rights 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
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01342185