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On the Completeness of Quantum Computation Models

Pablo J. Arrighi, Gilles Dowek B

Published Mar 20, 2010• 9 citations
OpenAlex

Abstract

Abstract. The notion of computability is stable (i.e. independent of the choice of an indexing) over infinite-dimensional vector spaces pro-vided they have a finite “tensorial dimension”. Such vector spaces with a finite tensorial dimension permit to define an absolute notion of com-pleteness for quantum computation models and give a precise meaning to the Church-Turing thesis in the framework of quantum theory. 1

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Abstract. The notion of computability is stable (i.e. independent of the choice of an indexing) over infinite-dimensional vector spaces pro-vided they have a finite “tensorial dimension”. Such vector spaces with a finite tensorial dimension permit to define an absolute notion of com-pleteness for quantum computation models and give a precise meaning to the Church-Turing thesis in the framework of quantum theory. 1
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].
It is possible in principle to construct quantum mechanical observables and unitary operators which, if implemented in physical systems as measurements and dynamical evolution, would contradict the Church-Turing thesis, which lies at the foundation of computer science. Elsewhere we have argued that the quantum measurement problem implies a finite, computational model of the measurement and evolution of quantum states. If correct, this approach helps to identify the key feature that can reconcile quantum mechanics with the Church-Turing thesis: finitude of the degree of fine-graining of Hilbert space. This suggests that the Church-Turing thesis constrains the physical universe and thereby highlights a surprising connection between purely logical and algorithmic considerations on the one hand and physical reality on the other.
According to the Church-Turing Thesis (CTT), effective formal behaviours can be simulated by Turing machines; this has naturally led to speculation that physical sys- tems 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 princi- ples, 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 compu- tational 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.
In the search for a precise definition of the notion of effective calculability, Turing’s computability has proven to be more convincing than other definitions proposed almost simultaneously. Turing does not define a particular class of “calculable functions”, but which processes can be carried out in computing. His conceptual analysis brings about an intimate and inescapable link between computability and “measurability”, hence between mechanical procedures and physical processes. The full significance of this link is captured in Deutsch’s physical version of the Church-Turing thesis, namely, his claim that every physical system can be simulated by a universal quantum Turing machine. Quantum physics supports more efficient forms of computation, even though quantum computation remains within Turing’s limits. Why? This chapter explores how logical principles and physical processes coalesce into the notion of computation.