Necessary and sufficient conditions for quantum computation

Necessary and sufficient conditions are given for a quantummechanical system to possess a coordinate system with respect to which its behaviour at discrete times is that of a universal digital computer . The form of the diagonal representation for the unitary time evolution operator for quantum universal computers is derived ; aspects of the transformation between the diagonal representation and the computational representation are shown to be uncomputable . A quantum-mechanical treatment of macroscopic, dissipative computers is given . 1 . Introduction When can a quantum-mechanical system compute? Digital computers are machines that can be programmed to perform logical and arithmetical operations . Contemporary digital computers are `universal', in the sense that a program that runs on one computer can, if properly compiled, run on any other computer that has access to enough memory space and time . Any one universal computer can simulate the operation of any other ; and the set of tasks that any such machine can perform is common to all universal machines . Many individual systems, ranging from cellular automata [1] to hard-sphere gases [2], have been shown to be capable of universal computation [3, 4] . Recently, Moore [5] has exhibited a class of classical nonlinear dynamical systems whose time evolutions when described in the proper system of coordinates are equivalent to the action of Turing machines, conceptual prototypes of the digital computer . Since Bennett's discovery [6] that computation can be carried out in a non-dissipative fashion, a number of Hamiltonian quantum-mechanical systems have been proposed whose time-evolutions over discrete intervals are equivalent to those of specific universal Turing machines [7-10] . The first quantum-mechanical treatment of computers was given by Benioff [7], who exhibited a Hamiltonian system with a basis whose members corresponded to the logical states of a Turing machine, and whose unitary evolution transformed those basis states at integer times into the states corresponding to their logical successors. In order to make the Hamiltonian local, in the sense that its structure depended only on the part of the computation being performed at that time, Benioff found it necessary to make the Hamiltonian time-dependent . Feynman [8] discovered a way to make the computational Hamiltonian both local and time-independent by incorporating the direction of computation in the initial 0950-0340/94 $10 . 00 © 1994 Taylor & Francis Ltd .