Abstract
The theory of logical gates in quantum computation has inspired the development of new forms of quantum logic based on the following semantic idea: the meaning of a formula is identified with a quantum information quantity, represented by a density operator. At the same time, the logical connectives are interpreted as operations defined in terms of quantum gates. In this framework, some possible relations between fuzzy representations based on continuous t-norms for quantum gates and the probabilistic behavior of quantum computational finite-valued connectives are investigated. In particular, a fuzzy-type representation for quantum many-valued extensions of the gates introduced by Toffoli, Fredkin and Peres is described.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bertini C, Leporini R (2007) Quantum computational finite-valued logics. Int J Quantum Inf 5(5):641–665
Bloch F (1946) Nuclear induction. Phys Rev 70(7–8):460–474
Chang CC (1958) Algebraic analysis of many valued logics. Trans Am Math Soc 88:467
Chang CC (1959) A new proof of the completeness of Łukasiewicz axioms. Trans Am Math Soc 93:74
Dalla Chiara ML, Giuntini R, Leporini R (2005) Logics from quantum computation. Int J Quantum Inf 3:293–337
Dalla Chiara ML, Giuntini R, Leporini R (2006) Compositional and holistic quantum computational semantics. Nat Comput. https://doi.org/10.1007/s11047-006-9020-x
Dalla Chiara ML, Giuntini R, Freytes H, Ledda A, Sergioli G (2009) The algebraic structure of an approximately universal system of quantum computational gates. Found Phys 39:559–572
Dalla Chiara ML, Giuntini R, Leporini R, Sergioli G (2016) A many-valued approach to quantum computational logics. Fuzzy Sets Syst. https://doi.org/10.1016/j.fss.2016.12.015
Fredkin E, Toffoli T (1982) Conservative logic. Int J Theor Phys 21:219–253
Freytes H, Sergioli G, Aricò A (2010) Representing continuous t-norms in quantum computation with mixed states. J Phys A Math Theor 43:465306
Islam MS, Rahman MM, Begum Z, Hafiz MZ (2009) Low cost quantum realization of reversible multiplier circuit. Inf Technol J 8(2):208–213
Leporini R, Bertini C (2018) Quantum structures in qudit spaces. Theor Comput Sci 752:86–96
Petri C.A (1967) Gründsatzliches zur Beschreibung diskreter Prozesse. In: Proceedings of the Third Colloq. über Automatentheorie, Hannover, 1965. Birkhäuser Basel, p 121. English version: Fundamentals of the representation of discrete processes, ISF report 82.04, 1982. (trans: Genrich HJ, Thiagarajan PS)
Zawirski Z (1934) Relation of many-valued logic to probability calculus (in Polish, original title: stosunek logiki wielowarto sciowej do rachunku prawdopodobie nstwa). Prace Komisji Filozoficznej Pozna nskiego Towarzystwa Przyjaciół Nauk 4:155
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Ethical approval
This article does not contain any studies with human participants or animals performed by any of the authors.
Additional information
Communicated by Federico Holik.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Leporini, R., Bertini, C. & Fabiani, F.C. Fuzzy representation of finite-valued quantum gates. Soft Comput 24, 10305–10313 (2020). https://doi.org/10.1007/s00500-020-04870-3
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-020-04870-3