Universal Turing machine

Abstract: In this paper we study quantum computation from a complexity theoretic viewpoint. Our first result is the existence of an efficient universal quantum Turing machine in Deutsch's model of a quantum Turing machine (QTM) [Proc. Roy. Soc. London Ser. A, 400 (1985), pp. 97--117]. This construction is substantially more complicated than the corresponding construction for classical Turing machines (TMs); in fact, even simple primitives such as looping, branching, and composition are not straightforward in the context of quantum Turing machines. We establish how these familiar primitives can be implemented and introduce some new, purely quantum mechanical primitives, such as changing the computational basis and carrying out an arbitrary unitary transformation of polynomially bounded dimension. We also consider the precision to which the transition amplitudes of a quantum Turing machine need to be specified. We prove that $O(\log T)$ bits of precision suffice to support a $T$ step computation. This justifies the claim that the quantum Turing machine model should be regarded as a discrete model of computation and not an analog one. We give the first formal evidence that quantum Turing machines violate the modern (complexity theoretic) formulation of the Church--Turing thesis. We show the existence of a problem, relative to an oracle, that can be solved in polynomial time on a quantum Turing machine, but requires superpolynomial time on a bounded-error probabilistic Turing machine, and thus not in the class $\BPP$. The class $\BQP$ of languages that are efficiently decidable (with small error-probability) on a quantum Turing machine satisfies $\BPP \subseteq \BQP \subseteq \Ptime^{\SP}$. Therefore, there is no possibility of giving a mathematical proof that quantum Turing machines are more powerful than classical probabilistic Turing machines (in the unrelativized setting) unless there is a major breakthrough in complexity theory.

Abstract: In this dissertation we study quantum computation from a complexity theoretic viewpoint. Our first result is the existence of an efficient universal quantum Turing Machine in Deutsch's model of a quantum Turing Machine. This construction is substantially more complicated than the corresponding construction for classical Turing Machines--in fact, even simple primitives such as looping, branching and composition are not straightforward in the context of quantum Turing Machines. We establish how these familiar primitives can be implemented, and also introduce some new, purely quantum mechanical primitives, such as changing the computational basis, and carrying out an arbitrary unitary transformation of polynomially bounded dimension. We also consider the precision to which the transition amplitudes of a quantum Turing Machine need to be specified. We prove that O(log T) bits of precision suffice to support a T step computation. This justifies the claim that the quantum Turing Machine model should be regarded as a discrete model of computation and not an analog one. We give the first evidence indicating that quantum Turing Machines are more powerful than classical probabilistic Turing Machines. We show the existence of a problem, relative to an oracle, that can be solved in polynomial time on a quantum Turing Machine, but requires super-polynomial time on a bounded-error probabilistic Turing Machine; and thus not in the class BPP. In fact, we show that this problem cannot be solved in MA relative to the same oracle, thus showing that even non-determinism together with randomness is not sufficient to solve the problem in poly-nomial time. The class BQP, of languages that are efficiently decidable (with small error-probability) on a quantum Turing Machine, satisfies: BPP $\subseteq$ BQP $\subseteq$ P$\sp{\sharp P}$. Therefore there is no possibility of giving a mathematical proof that quantum Turing Machines are more powerful than classical probabilistic Turing Machines (in the unrelativized setting) unless there is a major breakthrough in complexity theory. We also give evidence suggesting that quantum Turing Machines cannot efficiently solve all of NP. Specifically, we prove that relative to an oracle chosen uniformly at random, with probability 1, the class NP cannot be solved on a quantum Turing machine in time $o(2\sp{n/2}).$

Abstract: In this paper a microscopic quantum mechanical model of computers as represented by Turing machines is constructed. It is shown that for each numberN and Turing machineQ there exists a HamiltonianH N Q and a class of appropriate initial states such that if c is such an initial state, thenψ Q N (t)=exp(−1H N Q t)ψ Q N (0) correctly describes at timest 3,t 6,⋯,t 3N model states that correspond to the completion of the first, second, ⋯, Nth computation step ofQ. The model parameters can be adjusted so that for an arbitrary time intervalΔ aroundt 3,t 6,⋯,t 3N, the “machine” part ofψ Q N (t) is stationary.

Abstract: The use of Turing machines for calculating finite binary sequences is studied from the point of view of information theory and the theory of recursive functions. Various results are obtained concerning the number of instructions in programs. A modified form of Turing machine is studied from the same point of view. An application to the problem of defining a patternless sequence is proposed in terms of the concepts here developed.