Reversibility and irreversibility in quantum computation and in quantum computational logics

  • Authors:

  • Maria Luisa Dalla Chiara;Roberto Giuntini;Roberto Leporini

  • Affiliations:

  • Dipartimento di Filosofia, Università di Firenze, Firenze, Italy;Dipartimento di Scienze Pedagogiche e Filosofiche, Università di Cagliari, Cagliari, Italy;Dipartimento di Matematica, Statistica, Informatica e Applicazioni, Università di Bergamo, Bergamo, Italy

  • Venue:
  • Algebraic and proof-theoretic aspects of non-classical logics
  • Year:
  • 2007

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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.