Partial Observation of Quantum Turing Machines and a Weaker Well-Formedness Condition

Simon Perdrix 1
1 LIG Laboratoire d'Informatique de Grenoble - CAPP
LIG - Laboratoire d'Informatique de Grenoble
Abstract : The quantum Turing machine (QTM) has been introduced by Deutsch as an abstract model of quantum computation. The transition function of a QTM is linear, and has to be unitary to be a well-formed QTM. This well-formedness condition ensures that the evolution of the machine does not violate the postulates of quantum mechanics. However, we claim in this paper that the well-formedness condition is too strong and we introduce a weaker condition, leading to a larger class of Turing machines called Observable Quantum Turing Machines (OQTMs). We prove that the evolution of such OQTM does not violate the postulates of quantum mechanics while offering a more general abstract model for quantum computing. This novel abstract model unifies classical and quantum computations, since every well-formed QTM and every deterministic TM are OQTMs, whereas a deterministic TM has to be reversible to be a well-formed QTM. In this paper we introduce the fundamentals of OQTM like a well-observed lemma and a completion lemma. The introduction of such an abstract machine allowing classical and quantum computations is motivated by the emergence of models of quantum computation like the one-way model. More generally, the OQTM aims to be an abstract framework for the pragmatic paradigm of quantum computing: quantum data, classical control'. Furthermore, this model allows a formal and rigorous treatment of problems requiring classical interactions, like the halting of QTM. Finally, it opens new perspectives for the construction of a universal QTM.
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Submitted on : Wednesday, January 15, 2014 - 11:49:04 AM
Last modification on : Friday, October 25, 2019 - 1:30:03 AM

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Simon Perdrix. Partial Observation of Quantum Turing Machines and a Weaker Well-Formedness Condition. Electronic Notes in Theoretical Computer Science, Elsevier, 2011, 270 (1), pp.99-111. ⟨10.1016/j.entcs.2011.01.009⟩. ⟨hal-00931367⟩

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