Unitary dilation models of Turing machines in quantum mechanics

Phys. Rev. A 51, 3513 – Published 1 May 1995
Paul Benioff

Abstract

A goal of quantum-mechanical models of the computation process is the description of operators that model changes in the information-bearing degrees of freedom. Iteration of the operators should correspond to steps in the computation, and the final state of halting computations should be stable under iteration. The problem is that operators constructed directly from the process description do not have these properties. In general these operators annihilate the halted state. If information-erasing steps are present, there are additional problems. These problems are illustrated in this paper by consideration of operators for two simple one-step processes and two simple Turing machines. In general the operators are not unitary and, if erasing steps are present, they are not even contraction operators. Various methods of extension or dilation to unitary operators are discussed. Here unitary power dilations are considered as a solution to these problems. It is seen that these dilations automatically provide a good solution to the initial- and final-state problems. For processes with erasing steps, recording steps must be included prior to the dilation, but only for the steps that erase information. Hamiltonians for these processes are also discussed. It is noted that H, described by exp(-iHΔ)=UT, where UT is a unitary step operator for the process and Δ a time interval, has complexity problems. These problems and those noted above are avoided here by the use of the Feynman approach to constructing Hamiltonians directly from the unitary power dilations of the model operators. It is seen that the Hamiltonians so constructed have some interesting properties.

DOI: http://dx.doi.org/10.1103/PhysRevA.51.3513

  • Received 6 January 1995
  • Published in the issue dated May 1995

© 1995 The American Physical Society

Authors & Affiliations

Paul Benioff

  • Environmental Assessment Division, Building 900, Argonne National Laboratory, Argonne, Illinois 60439

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