Abstract
We study the possibility of performing fuzzy set operations on a quantum computer. After giving a brief overview of the necessary quantum computational and fuzzy set theoretical concepts we demonstrate how to encode the membership function of a digitized fuzzy number in the state space of a quantum register by using a suitable superposition of tensor product states that form a computational basis. We show that a standard quantum adder is capable to perform Kaufmann's addition of fuzzy numbers in the course of only one run by acting at once on all states in the superposition, which leads to a considerable gain in the number of required operations with respect to performing such addition on a classical computer.
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REFERENCES
Dubois, D. and Prade, H. (1980). Fuzzy Sets and Systems. Theory and Applications, Academic Press, New York.
Ekert, A. and Jozsa, R. (1996). Quantum computation and Shor's factoring algorithm. Reviews of Modern Physics 68, 733–753.
Kaufmann, A. (1975). Introduction to the Theory of Fuzzy Subsets, Vol. I, Academic Press, New York.
Pykacz, J. (1987). Quantum logics as families of fuzzy subsets of the set of physical states. In Preprints of the Second IFSA Congress, Tokyo, Vol. 2, International Fuzzy Systems Association, Japan Chapter, Tokyo, 1987, pp. 437–440.
Pykacz, J. (1994). Fuzzy quantum logics and infinite-valued L ukasiewicz logic. International Journal of Theoretical Physics 33, 1403–1416.
Pykacz, J. (1998). Fuzzy quantum logics as a basis for quantum probability theory, International Journal of Theoretical Physics 37, 281–290.
Pykacz, J. (2000a). L ukasiewicz operations in fuzzy set and many-valued representation of quantum logics. Foundations of Physics 37, 1503–1524.
Pykacz, J. (2000b). Quantum logic as a basis for computations. International Journal of Theoretical Physics 39, 839–850.
Shor, P. W. (1994). Algorithms for quantum computation: Discrete logarithms and factoring. In Proceedings of the 35th Annual Symposium on the Foundations of Computer Science, S. Goldwasser, ed., IEEE Computer Society Press, Los Alamitos, CA, pp. 124–134.
Vedral, V., Barenco, A., and Ekert, A. (1996). Quantum networks for elementary arithmetic operations, Physical Review A 54, 147–153.
Vedral, V. and Plenio, M. B. (1998). Basics of quantum computation. Progress of Quantum Electronics 22,1–40; see also e-print: quant-ph/9802065.
Zalka, C. (1998). Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London Series A 454, 313–32; see also e-print: quant-ph/9603026.
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D'hooghe, B., Pykacz, J. & Zapatrin, R.R. Quantum Computation of Fuzzy Numbers. International Journal of Theoretical Physics 43, 1423–1432 (2004). https://doi.org/10.1023/B:IJTP.0000048625.57350.4f
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DOI: https://doi.org/10.1023/B:IJTP.0000048625.57350.4f