Quantum theory , the Church-Turing principle and the universal quantum computer

It is argued that underlying the Church-Turing hypothesis there is an implicit physical assertion. Here, this assertion is presented explicitly as a physical principle: ‘every finitely realizable physical system can be perfectly simulated by a universal model computing machine operating by finite means’. Classical physics and the universal Turing machine, because the former is continuous and the latter discrete, do not obey the principle, at least in the strong form above. A class of model computing machines that is the quantum generalization of the class of Turing machines is described, and it is shown that quantum theory and the ‘universal quantum computer’ are compatible with the principle. Computing machines resembling the universal quantum computer could, in principle, be built and would have many remarkable properties not reproducible by any Turing machine. These do not include the computation of non-recursive functions, but they do include ‘quantum parallelism’, a method by which certain probabilistic tasks can be performed faster by a universal quantum computer than by any classical restriction of it. The intuitive explanation of these properties places an intolerable strain on all interpretations of quantum theory other than Everett’s. Some of the numerous connections between the quantum theory of computation and the rest of physics are explored. Quantum complexity theory allows a physically more reasonable definition of the ‘complexity’ or ‘knowledge’ in a physical system than does classical complexity theory. Current address: Centre for Quantum Computation, Clarendon Laboratory, Department of Physics, Parks Road, OX1 3PU Oxford, United Kingdom.  david.deutsch@qubit.org This version (Summer 1999) was edited and converted to L TEX by Wim van Dam at the Centre for Quantum Computation. Email:wimvdam@qubit.org 1 Computing machines and the Church-Turing principle The theory of computing machines has been extensively developed during the last few decades. Intuitively, a computing machine is any physical system whose dynamical evolution takes it from one of a set of ‘input’ states to one of a set of ‘output’ states. The states are labelled in some canonical way, the machine is prepared in a state with a given input label and then, following some motion, the output state is measured. For a classical deterministic system the measured output label is a definite functionf of the prepared input label; moreover the value of that label can in principle be measured by an outside observer (the ‘user’) and the machine is said to ‘compute’the functionf . Two classical deterministic computing machines are ‘computationally equivalent’ under given labellings of their input and output states if they compute the same function under those labellings. But quantum computing machines, and indeed classical stochastic computing machines, do not ‘compute functions’ in the above sense: the output state of a stochastic machine is random with only the probability distribution function for the possible outputs depending on the input state. The output state of a quantum machine, although fully determined by the input state is not an observable and so the user cannot in general discover its label. Nevertheless, the notion of computational equivalence can be generalized to apply to such machines also. Again we define computational equivalence under given labellings,but it is now necessary to specify more precisely what is to be labelled. As far as the input is concerned, labels must be given for each of the possible ways of preparing the machine, which correspond, by definition, to all the possible input states. This is identical with the classical deterministic case. However, there is an asymmetry between input and output because there is an asymmetry between preparation and measurement: whereas a quantum system can be prepared in any desired permitted input state, measurement cannot in general determine its output state; instead one must measure the value of some observable. (Throughout this paper I shall be using the Schr ̈ odinger picture, in which the quantum state is a function of time but observables are constant operators.) Thus what must be labelled is the set of ordered pairs consisting of an output observable and a possible measured value of that observable (in quantum theory, a Hermitian operator and one of its eigenvalues). Such an ordered pair contains, in effect, the specification of a possible experiment that could be made on the output, together with a possible result of that experiment. Two computing machines are computationally equivalent under given labellings if in any possible experiment or sequence of experiments in which their inputs were prepared equivalently under the input labellings, and observables corresponding to each other under the output labellings were measured, the measured values of these observables for the two machines would be statistically indistinguishable. That is, the probability distribution functions for the outputs of the two machines would be identical. In the sense just described, a given computing machine M computes at most one function. However, there ought to be no fundamental difference between altering the input state in which M is prepared, and altering systematically the constitution of M so that it becomes a different machine M0 computing a different function. To formalize such operations, it is often useful to consider machines with two inputs, the preparation of one constituting a ‘program’ determining which function of the other is to be computed. To each such machine M there corresponds a set C (M) of ‘M-computable functions’. A functionf isM-computable ifM can computef when prepared with some program. The set C(M) can be enlarged by enlarging the set of changes in the constitution of M that are labelled as possible M-programs. Given two machines M andM0 it is possible to construct a composite machine whose set of computable functions contains the union of C (M) and C(M0). There is no purely logical reason why one could not go on ad infinitumbuilding more powerful