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On the Halting Problem for Quantum Turing Machines

Masanao Ozawa

Published Oct 1, 1998• 2 citations
OpenAlex

Abstract

Foundations of the theory of quantum Turing machines are investigated.The local transition functions are characterized for fully general quantum Tur- ing machines.Foundational problems on the preparation and the measurement of quantum Turing machines are discussed.The halt scheme, originally proposed by Deutsch, is formulated precisely and is shown to work without spoiling computation.

Related papers

Quantum Turing Machines: Local Transition, Preparation, Measurement, and Halting

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Foundations of the theory of quantum Turing machines are investigated. The protocol for the preparation and the measurement of quantum Turing machines is discussed. The local transition functions are characterized for fully general quantum Turing machines. A new halting protocol is proposed without augmenting the halting qubit and is shown to work without spoiling the computation.

arXiv:quant-ph/9809038•Kluwer Academic Publishers eBooks•Dec 19, 2005•11 citations

Local Transition Functions of Quantum Turing Machines

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Foundations of the notion of quantum Turing machines are investigated. According to Deutsch's formulation, the time evolution of a quantum Turing machine is to be determined by the local transition function. In this paper, the local transition functions are characterized for fully general quantum Turing machines, including multi-tape quantum Turing machines, extending the results due to Bernstein and Vazirani.

arXiv:quant-ph/9811069•Nov 25, 1998•1 citation

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We study the observation of quantum Turing machines by allowing interactions between a quantum machine and its environment during the computation, whereas a quantum Turing machine --original model introduced by Deutsch-- remains isolated. We show that the introduction of observations leads to a weakening of the well formedness conditions of quantum Turing machines such that any (reversible or not) classical Turing machine is a special instance of a quantum machine. Moreover, observation of quantum Turing machines provides a formal solution to the halting process problem: the impossibility to know whether a given quantum Turing machine has actually reached its halting state. It also provides a more realistic abstract architecture of a quantum computer while most of the physical proposals of quantum computers are based on an hybrid classical-quantum architecture. However, we show that a natural formalisation of an observable quantum Turing machines leads to an over-powerful model solving undecidable problems. As a consequence, we introduce a restricted version of observable quantum Turing machines and we show that, under this restriction, any observable quantum Turing machine can be efficiently simulated by a quantum Turing machine. Finally, we discuss the potential application of the observable quantum Turing machine in the quest of a universal quantum Turing machine while recent papers have pointed out that a classical control is a key feature of a universal quantum machine.

•Jan 1, 2011•0 citations

Quantum Nondemolition Monitoring of Universal Quantum Computers

Masanao Ozawa

The halt scheme for quantum Turing machines, originally proposed by Deutsch, is reformulated precisely and is proven to work without spoiling the computation. The ``conflict'' pointed out recently by Myers in the definition of a universal quantum computer is shown to be only apparent. In the context of quantum nondemolition (QND) measurement, it is also shown that the output observable, an observable representing the output of the computation, is a QND observable and that the halt scheme is equivalent to the QND monitoring of the output observable.

arXiv:quant-ph/9704028•Physical Review Letters•Jan 19, 1998•45 citations
Foundations of the theory of quantum Turing machines are investigated.The local transition functions are characterized for fully general quantum Tur- ing machines.Foundational problems on the preparation and the measurement of quantum Turing machines are discussed.The halt scheme, originally proposed by Deutsch, is formulated precisely and is shown to work without spoiling computation.
We study the observation of quantum Turing machines by allowing interactions between a quantum machine and its environment during the computation, whereas a quantum Turing machine --original model introduced by Deutsch-- remains isolated. We show that the introduction of observations leads to a weakening of the well formedness conditions of quantum Turing machines such that any (reversible or not) classical Turing machine is a special instance of a quantum machine. Moreover, observation of quantum Turing machines provides a formal solution to the halting process problem: the impossibility to know whether a given quantum Turing machine has actually reached its halting state. It also provides a more realistic abstract architecture of a quantum computer while most of the physical proposals of quantum computers are based on an hybrid classical-quantum architecture. However, we show that a natural formalisation of an observable quantum Turing machines leads to an over-powerful model solving undecidable problems. As a consequence, we introduce a restricted version of observable quantum Turing machines and we show that, under this restriction, any observable quantum Turing machine can be efficiently simulated by a quantum Turing machine. Finally, we discuss the potential application of the observable quantum Turing machine in the quest of a universal quantum Turing machine while recent papers have pointed out that a classical control is a key feature of a universal quantum machine.
The halt scheme for quantum Turing machines, originally proposed by Deutsch, is reformulated precisely and is proven to work without spoiling the computation. The ``conflict'' pointed out recently by Myers in the definition of a universal quantum computer is shown to be only apparent. In the context of quantum nondemolition (QND) measurement, it is also shown that the output observable, an observable representing the output of the computation, is a QND observable and that the halt scheme is equivalent to the QND monitoring of the output observable.