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Cambridge University Press eBooks

Theoretical Computer Science

Noson S. Yanofsky, Mirco A. Mannucci

Published Aug 11, 2008• 22 citations
Publisher

Abstract

The meaning of the world is the separation of wish and fact. Kurt Gödel, quoted in Hao Wang's A Logical Journey: From Gödel to Philosophy , page 309 In a sense, theoretical computer science is uniquely qualified to study quantum computing. After all, Alan Turing and the other founders of theoretical computer science studied formal computation long before engineers actually produced a real-life computer. At present, large-scale quantum computers are not a reality yet. Nevertheless, the theoretical analysis of quantum computability and complexity is well on its way. In Section 8.1, we start with a quick review of some of the basics of deterministic and nondeterministic Turing machines and the complexity classes that they engender. However, we shall discuss them in a way that is easily generalizable for our purposes. Section 8.2 moves on to probabilistic Turing machines and their zoo of complexity classes. Our main objective is found in Section 8.3, where we meet quantum Turing machines and their complexity classes. We shall also state some basic theorems and ideas about quantum computation. DETERMINISTIC AND NONDETERMINISTIC COMPUTATIONS Theoretical computer science deals with the question, “What is computable?” We must immediately qualify the question: “computable according to which model of computation?” It turns out that if we omit the question of efficiency, all sufficiently complicated formal models of computation can simulate each other. However, in order to fix our ideas and notation, we have to stick with one and work with it.

DOI
10.1017/cbo9780511813887.010

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Cambridge University Press eBooks•May 1, 2014•0 citations
The meaning of the world is the separation of wish and fact. Kurt Gödel, quoted in Hao Wang's A Logical Journey: From Gödel to Philosophy , page 309 In a sense, theoretical computer science is uniquely qualified to study quantum computing. After all, Alan Turing and the other founders of theoretical computer science studied formal computation long before engineers actually produced a real-life computer. At present, large-scale quantum computers are not a reality yet. Nevertheless, the theoretical analysis of quantum computability and complexity is well on its way. In Section 8.1, we start with a quick review of some of the basics of deterministic and nondeterministic Turing machines and the complexity classes that they engender. However, we shall discuss them in a way that is easily generalizable for our purposes. Section 8.2 moves on to probabilistic Turing machines and their zoo of complexity classes. Our main objective is found in Section 8.3, where we meet quantum Turing machines and their complexity classes. We shall also state some basic theorems and ideas about quantum computation. DETERMINISTIC AND NONDETERMINISTIC COMPUTATIONS Theoretical computer science deals with the question, “What is computable?” We must immediately qualify the question: “computable according to which model of computation?” It turns out that if we omit the question of efficiency, all sufficiently complicated formal models of computation can simulate each other. However, in order to fix our ideas and notation, we have to stick with one and work with it.
A practical viewpoint links reality, representation, and language to calculation by the concept of Turing (1936) machine being the mathematical model of our computers. After the Gödel incompleteness theorems (1931) or the insolvability of the so-called halting problem (Turing 1936; Church 1936) as to a classical machine of Turing, one of the simplest hypotheses is completeness to be suggested for two ones. That is consistent with the provability of completeness by means of two independent Peano arithmetics discussed in Section I.Many modifications of Turing machines cum quantum ones are researched in Section II for the Halting problem and completeness, and the model of two independent Turing machines seems to generalize them.Then, that pair can be postulated as the formal definition of reality therefore being complete unlike any of them standalone, remaining incomplete without its complementary counterpart. Representation is formal defined as a one-to-one mapping between the two Turing machines, and the set of all those mappings can be considered as “language” therefore including metaphors as mappings different than representation. Section III investigates that formal relation of “reality”, “representation”, and “language” modeled by (at least two) Turing machines.The independence of (two) Turing machines is interpreted by means of game theory and especially of the Nash equilibrium in Section IV.Choice and information as the quantity of choices are involved. That approach seems to be equivalent to that based on set theory and the concept of actual infinity in mathematics and allowing of practical implementations.
The physics and the mathematics of computation are examined to provide a foundation and perspective for the investigation of the quantum mechanics of computation. Our purpose is to explore the fundamental limits and constraints imposed on computation by Nature through the laws of physics and the mathematics of computational complexity. Inasmuch as information storage and transmission are an integral part of computation, their physical bounds are considered. The computer is viewed both physically and mathematically as a dynamical system, and is depicted in terms of the basic Turing machine paradigm. Three fundamental classes of the Turing machine are defined; the deterministic, stochastic and quantum Turing machines. Hamiltonian models and physical realizations of quantum computing are described. Quantum computers can perform some tasks which have no classical analogue, but they cannot compute functions that are non-computable by classical means. Some classically intractable problems can be solved with quantum computers.
There is strong evidence that computers based upon the principles of quantum physics represent an inherently new and more powerful model of computation. Such computers violate the modern form of the Church-Turing thesis (which lies at the foundations of computer science). This thesis can be informally summarized as follows: all physical implementations of computing devices can be described by the same abstract model of computation-the probabilistic Turing Machine or the equivalent random access machine. In this paper, we will describe two formal models for quantum computers: quantum circuits and quantum Turing Machines, introduced by Deutsch [1985].
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)O(\log T)O(logT) 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.
. We revisit the notion of a quantum Turing-machine , whose design is based on the laws of quantum mechanics. It turns out that such a machine is not more powerful, in the sense of computability, than the machine originally constructed by Turing. Quantum Turing-machines do not violate the Church–Turing thesis. The benefit of quantum computing lies in efficiency . Quantum computers appear to be more efficient, in time, than classical Turing-machines, however its exact additional computational power is unclear, as this question ties in with deep open problems in complexity theory. We will sketch where BQP, the quantum analogue of the complexity class P, resides in the realm of complexity classes. §1. Introduction . A decade before Turing developed his theory of computing, physicists struggled with the advent of quantum mechanics. During the famous 5th Solvay Conference in 1927 it was clear that a new era of physics had surfaced. Its strange features like superposition and entanglement still lead to heated discussions and much confusion. However strange and counter-intuitive, the theory has never been refuted by experiments that are performed daily and in great numbers throughout laboratories around the world. Time after time the predictions of quantum mechanics are in full agreement with experiment. Shortly after the advent of quantum mechanics, Church, Turing and Post developed the notion of computability [Chu36, Tur36, Pos36]. Less than 10 years later these formal ideas would be put to practice resulting in the ENIAC, the first general purpose machine.