Quantum Mechanical Hamiltonian Models of Computers a

Interest in the physical limitations of the computation process has been increasing in recent years.’.’ Landauer3“ has discussed this subject extensively, particularly from the viewpoint of energy dissipation and reversible and irreversible computation. He has also stressed the fact that the computation process is necessarily implemented as a physical process and is thus subject to physical laws. Much of the work in this area has been concerned with the construction of various models of the computation process with different properties. Bennett’s8 and Likharev9 have discussed thermodynamically reversible models that dissipate arbitrarily little energy if they proceed sufficiently slowly. Fredkin and Toffoli” have constructed ideal classical mechanical billiard-ball models of the computation process that do not dissipate energy and that evolve at a finite speed. Benioff has constructed quantum mechanical Hamiltonian models of Turing machines on spin lattices both and w i t h o ~ t ’ ~ ? ’ ~ the use of scattering systems to drive the process. Feynman” has outlined the construction of quantum ballistic models of the computation process. He has also shown that models can be constructed for which the Hamiltonian contains, at most, three body interactions. Margolusi6 has used Feynman’s methods to construct quantum mechanical models of the classical billiard-ball rn~dels . ’~ Peres” has constructed models that include redundancy to correct errors, and Zurek” has discussed the propagation of errors in classical and quantum mechanical models. Deutsch” and Pitowsky” have considered the computation power of more general types of quantum computers. In this paper, Feynrnan’s’’ methods will be used to construct quantum mechanical Hamiltonian models of Turing machines on a two-dimensional lattice of spin+ systems. The Hamiltonians will be seen to be spatially local, simple, and such that the system state does not degrade as it evolves. From the point of view of constructing physical models of these machines, these properties are desirable. Physically reasonable Hamiltonians are spatially local in that their interaction does not extend over all space with undiminished strength. A Hamiltonian is simple if its construction requires, at most, knowledge of the computer programs to be run. This is reasonable in that real computers require this knowledge for their operation. Knowledge of computation orbits is not needed. Evolution without