Journal of Statistical Physics, FoL 29, No. 3, 1982 Quantum Mechanical Hamiltonian Models of Turing Machines Paul Benioff I Received October 5, 1981; rev&ed June 9, 1982 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). Succes- sive 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/computationspeed 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. KEY WORDS: Schr6dinger equation description of Turing machines; nondissipative models of computers; quantum spin lattices, 1. INTRODUCTION In recent years there has been an upsurge of interest in the physical limitations of the computation process. In particular the energy cost of computation or information transfer and whether or not there must be energy dissipation are the subjects of much discussion. (1-1~ Some years ago it was felt (3'7) that there must be dissipation associated with the computa- tion process because the process is irreversible. However, in 1973, Ben- n e t t (2~ constructed reversible models of the computation process and 1 Division of Environmental Impact Studies, Argonne National Laboratory, Argonne, Illinois 60439. 515 0022-4715/82/1100-0515503.00/0 9 1982 Plenum Publishing Corporation