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Universality in Quantum Computation

David Deutsch, Adriano Barenco, Artur Ekert

Published Jan 1, 1995• 309 citations
OpenAlex

Abstract

We show that in quantum computation almost every gate that operates on two or more bits is a universal gate. We discuss various physical considerations bearing on the proper definition of universality for computational components such as logic gates. Introduction It has been known for several years that the theory of quantum computers --- i.e. machines that rely on characteristically quantum phenomena to perform computations [1] --- is substantially different from the classical theory of computation, which is essentially the theory of the universal Turing machine. We may identify three important differences. Firstly, the properties of quantum computers are not postulated in abstracto but are deduced entirely from the laws of physics. Logically, this was already true of the classical theory, as Landauer [2] has pointed out, but the intuitive nature of the classically-available computational operations, and the millennia-long history of their study, allowed pioneers such as Turing, Chu...

Related papers

Universality in quantum computation

David Elieser Deutsch, Adriano Barenco, Artur Ekert

Abstract We show that in quantum computation almost every gate that operates on two or more bits is a universal gate. We discuss various physical considerations bearing on the proper definition of universality for computational components such as logic gates.

arXiv:quant-ph/9505018•Proceedings of the Royal Society of London Series A Mathematical and Physical Sciences•Jun 8, 1995•346 citations

Deutsch's Universal Quantum Turing Machine (Revisited)

Willem L. Fouché, Johannes Heidema, Jones, Glyn +1 more

Deutsch, Feynman, and Manin viewed quantum computing as a kind of universal physical simulation procedure. Much of the writing about quantum Turing machines has shown how these machines can simulate an arbitrary unitary transformation on a finite number of qubits. This interesting problem has been addressed most famously in a paper by Deutsch, and later by Bernstein and Vazirani. Quantum Turing machines form a class closely related to deterministic and probabilistic Turing machines and one might hope to find a universal machine in this class. A universal machine is the basis of a notion of programmability. The extent to which universality has in fact been established by the pioneers in the field is examined and a key notion in theoretical computer science (universality) is scrutinised. In a forthcoming paper, the authors will also consider universality in the quantum gate model.

arXiv:quant-ph/0701108•Jan 16, 2007•4 citations

Toward a Universal Quantum Computer

Yu Shi

Given the existence of a universal quantum Turing machine, there is no conflict in the definition of a universal quantum computer between being universal and the need for a halt qubit, as previously claimed. It is also pointed out that there exists a universal, i.e. programmable, quantum circuit which can perform an arbitrary quantum computation, by repeating the operation of a fixed gate array.

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A Theory of Computation Based on Quantum Logic (I)

Mingsheng Ying

The (meta)logic underlying classical theory of computation is Boolean (two-valued) logic. Quantum logic was proposed by Birkhoff and von Neumann as a logic of quantum mechanics more than sixty years ago. The major difference between Boolean logic and quantum logic is that the latter does not enjoy distributivity in general. The rapid development of quantum computation in recent years stimulates us to establish a theory of computation based on quantum logic. The present paper is the first step toward such a new theory and it focuses on the simplest models of computation, namely finite automata. It is found that the universal validity of many properties of automata depend heavily upon the distributivity of the underlying logic. This indicates that these properties does not universally hold in the realm of quantum logic. On the other hand, we show that a local validity of them can be recovered by imposing a certain commutativity to the (atomic) statements about the automata under consideration. This reveals an essential difference between the classical theory of computation and the computation theory based on quantum logic.

arXiv:cs/0403041•Mar 29, 2004•0 citations

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

David Deutsch

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.

Proceedings of the Royal Society of London A Mathematical and Physical Sciences•Jul 8, 1985•4,597 citations

Construction of a universal quantum computer

Antonio A. Lagana, Max A. Lohe, Lorenz von Smekal

We construct a universal quantum computer following Deutsch's original proposal of a universal quantum Turing machine (UQTM). Like Deutsch's UQTM, our machine can emulate any classical Turing machine and can execute any algorithm that can be implemented in the quantum gate array framework but under the control of a quantum program, and hence is universal. We present the architecture of the machine, which consists of a memory tape and a processor and describe the observables that comprise the registers of the processor and the instruction set, which includes a set of operations that can approximate any unitary operation to any desired accuracy and hence is quantum computationally universal. We present the unitary evolution operators that act on the machine to achieve universal computation and discuss each of them in detail and specify and discuss explicit program halting and concatenation schemes. We define and describe a set of primitive programs in order to demonstrate the universal nature of the machine. These primitive programs facilitate the implementation of more complex algorithms and we demonstrate their use by presenting a program that computes the NAND function, thereby also showing that the machine can compute any classically computable function.

Physical Review A•May 18, 2009•9 citations
We show that in quantum computation almost every gate that operates on two or more bits is a universal gate. We discuss various physical considerations bearing on the proper definition of universality for computational components such as logic gates. Introduction It has been known for several years that the theory of quantum computers --- i.e. machines that rely on characteristically quantum phenomena to perform computations [1] --- is substantially different from the classical theory of computation, which is essentially the theory of the universal Turing machine. We may identify three important differences. Firstly, the properties of quantum computers are not postulated in abstracto but are deduced entirely from the laws of physics. Logically, this was already true of the classical theory, as Landauer [2] has pointed out, but the intuitive nature of the classically-available computational operations, and the millennia-long history of their study, allowed pioneers such as Turing, Chu...
Deutsch, Feynman, and Manin viewed quantum computing as a kind of universal physical simulation procedure. Much of the writing about quantum Turing machines has shown how these machines can simulate an arbitrary unitary transformation on a finite number of qubits. This interesting problem has been addressed most famously in a paper by Deutsch, and later by Bernstein and Vazirani. Quantum Turing machines form a class closely related to deterministic and probabilistic Turing machines and one might hope to find a universal machine in this class. A universal machine is the basis of a notion of programmability. The extent to which universality has in fact been established by the pioneers in the field is examined and a key notion in theoretical computer science (universality) is scrutinised. In a forthcoming paper, the authors will also consider universality in the quantum gate model.
The (meta)logic underlying classical theory of computation is Boolean (two-valued) logic. Quantum logic was proposed by Birkhoff and von Neumann as a logic of quantum mechanics more than sixty years ago. The major difference between Boolean logic and quantum logic is that the latter does not enjoy distributivity in general. The rapid development of quantum computation in recent years stimulates us to establish a theory of computation based on quantum logic. The present paper is the first step toward such a new theory and it focuses on the simplest models of computation, namely finite automata. It is found that the universal validity of many properties of automata depend heavily upon the distributivity of the underlying logic. This indicates that these properties does not universally hold in the realm of quantum logic. On the other hand, we show that a local validity of them can be recovered by imposing a certain commutativity to the (atomic) statements about the automata under consideration. This reveals an essential difference between the classical theory of computation and the computation theory based on quantum logic.
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.
We construct a universal quantum computer following Deutsch's original proposal of a universal quantum Turing machine (UQTM). Like Deutsch's UQTM, our machine can emulate any classical Turing machine and can execute any algorithm that can be implemented in the quantum gate array framework but under the control of a quantum program, and hence is universal. We present the architecture of the machine, which consists of a memory tape and a processor and describe the observables that comprise the registers of the processor and the instruction set, which includes a set of operations that can approximate any unitary operation to any desired accuracy and hence is quantum computationally universal. We present the unitary evolution operators that act on the machine to achieve universal computation and discuss each of them in detail and specify and discuss explicit program halting and concatenation schemes. We define and describe a set of primitive programs in order to demonstrate the universal nature of the machine. These primitive programs facilitate the implementation of more complex algorithms and we demonstrate their use by presenting a program that computes the NAND function, thereby also showing that the machine can compute any classically computable function.