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Oxford University Press eBooks

Quantum Computation and the Church–Turing Hypothesis

Christopher G. Timpson

Published Apr 25, 2013• 0 citations
Publisher

Abstract

Some of the philosophical questions raised by the theory of quantum computation are discussed. First it is considered whether the possibility of exponential speed-up in quantum computation provides an argument for a more substantive notion of quantum information than so far allowed. It is concluded that this is not so. Then some questions regarding the status of the Church-Turing hypothesis in the light of quantum computation are considered. In particular, Deutsch’s claim that a physical principle, the Turing Principle, underlies the Church-Turing hypothesis is rebutted. Finally, the question of whether the Church-Turing hypothesis might serve as a constraint on the laws of physics is briefly considered.

DOI
10.1093/acprof:oso/9780199296460.003.0006

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Some of the philosophical questions raised by the theory of quantum computation are discussed. First it is considered whether the possibility of exponential speed-up in quantum computation provides an argument for a more substantive notion of quantum information than so far allowed. It is concluded that this is not so. Then some questions regarding the status of the Church-Turing hypothesis in the light of quantum computation are considered. In particular, Deutsch’s claim that a physical principle, the Turing Principle, underlies the Church-Turing hypothesis is rebutted. Finally, the question of whether the Church-Turing hypothesis might serve as a constraint on the laws of physics is briefly considered.
Abstract 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 realizible 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.
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 the paper we interpret the Church Turing thesis of Computation Algorithm and how it differs from the strong Church Turing thesis in the field of computation. Church Turing thesis can be classified as Strong Church Turing thesis and Normal Church Turing thesis. Church Turing hypothesis says “that a computing problem can be solved in any computer if and only if it can be solved on a very simple ‘machine’, named TURING MACHINE”. We introspect the difference in the model of computers that are developed on these two competitive algorithms. We will also be studying the physical realization of bits in a quantum computer with special mention to the commercial quantum computer systems offered at the moment. Our objective is to find out the relation of a quantum bit and the Strong Church Turing Thesis converging the gap between theoretical model and physical structure.
As was emphasized by Deutsch, quantum computation shatters complexity theory, but is innocuous to computability theory. Yet Nielsen and others have shown how quantum theory as it stands could breach the physical Church-Turing thesis. We draw a clear line as to when this is the case, in a way that is inspired by Gandy. Gandy formulates postulates about physics, such as homogeneity of space and time, bounded density and velocity of information — and proves that the physical Church-Turing thesis is a consequence of these postulates. We provide a quantum version of the theorem. Thus this approach exhibits a formal non-trivial interplay between theoretical physics symmetries and computability assumptions.
. 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.
According to the Church-Turing Thesis (CTT), effective formal behaviours can be simulated by Turing machines; this has naturally led to speculation that physical systems can also be simulated computationally. But is this wider claim true, or do behaviours exist which are strictly hypercomputational? Several idealised computational models are known which suggest the possibility of hypercomputation, some Newtonian, some based on cosmology, some on quantum theory. While these models' physicality is debatable, they nonetheless throw into question the validity of extending CTT to include all physical systems. We consider the physicality of hypercomputational behaviour from first principles, by showing that quantum theory can be reformulated in a way that explains why physical behaviours can be regarded as 'computing something' in the standard computational state-machine sense. While this does not rule out the physicality of hypercomputation, it strongly limits the forms it can take. Our model also has physical consequences; in particular, the continuity of motion and arrow of time become theorems within the basic model.