Abstract
A characteristic feature of quantum computation is the use of reversible logical operations. These correspond to quantum logical gates that are mathematically represented by unitary operators defined on convenient Hilbert spaces. Two questions arise: 1) to what extent is quantum computation bound to the use of reversible logical operations? 2) How to identify the logical operations that admit a quantum computational simulation by means of appropriate gates? We introduce the notion of quantum computational simulation of a binary function defined on the real interval [0,1], and we prove that for any binary Boolean function there exists a unique fuzzy extension admitting a quantum computational simulation. As a consequence, the Łukasiewicz conjunction and disjunction do not admit a quantum computational simulation.
We warmly thank Francesco Paoli for his stimulating comments on the paper.
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References
Birkhoff, G., von Neumann, J.: The logic of quantum mechanics. Annals of Mathematics 37, 823–843 (1936)
Dalla Chiara, M.L., Giuntini, R., Leporini, R.: Quantum Computational Logics. A Survey. In: Hendricks, V., Malinowski, J. (eds.) Trends in Logic. 50 Years of Studia Logica, pp. 229–271. Kluwer, Dordrecht (2003)
Dalla Chiara, M.L., Giuntini, R., Leporini, R.: Quantum computational logics and Fock space semantics. International Journal of Quantum Information 2, 1–8 (2004)
Dalla Chiara, M.L., Giuntini, R., Leporini, R.: Logics from quantum computation. International Journal of Quantum Information 3, 293–337 (2005)
Dalla Chiara, M.L., Giuntini, R., Leporini, R.: A holistic quantum computational semantics. Natural Computing 5, 1–20 (2006)
Dalla Chiara, M.L., Giuntini, R.: Quantum logics. In: Gabbay, G., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. VI, pp. 129–228. Kluwer, Dordrecht (2002)
Dalla Chiara, M.L., Giuntini, R., Greechie, R.: Reasoning in Quantum Theory. Kluwer, Dordrecht (2004)
Deutsch, D., Ekert, A., Lupacchini, R.: Machines, logic and quantum physics. Bulletin of Symbolic Logic 3, 265–283 (2000)
Gudder, S.: Quantum computational logic. International Journal of Theoretical Physics 42, 39–47 (2003)
Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge Univ. Press, Cambridge (2000)
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Dalla Chiara, M.L., Giuntini, R., Leporini, R. (2007). Reversibility and Irreversibility in Quantum Computation and in Quantum Computational Logics. In: Aguzzoli, S., Ciabattoni, A., Gerla, B., Manara, C., Marra, V. (eds) Algebraic and Proof-theoretic Aspects of Non-classical Logics. Lecture Notes in Computer Science(), vol 4460. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75939-3_6
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DOI: https://doi.org/10.1007/978-3-540-75939-3_6
Publisher Name: Springer, Berlin, Heidelberg
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