Paper:

# Emulating Qubits with Fuzzy Logic

## M. Skander Hannachi, Yutaka Hatakeyama, and Kaoru Hirota

Department of Computational Intelligence and Systems Science, Interdisciplinary Graduate School of Science and Engineering, Tokyo Institute of Technology, G3-49, 4259 Nagatsuta, Midori-ku, Yokohama 226-8502, Japan

An approach for emulating quantum circuits using conventional analog hardware is presented based on the intuitive similarity between fuzzy logic and quantum superposition, as well as some geometrical analogies. This approach has the advantage of being easy to implement on dedicated hardware for parallel processing of membership functions and fuzzy inference, compared to conventional quantum computing, which requires quantum mechanical systems which are extremely sensitive to noise and difficult to extend to large scale systems. Using geometrical analogies and a suitable transformation, qubits are modeled as pairs of fuzzy membership functions evolving on the unit square and basic one qubit gates are modeled as transformations on this unit square. A fuzzy implementation of the one bit and two bit Deutsch-Jozsa algorithm is proposed. Physical implementation and advantages, such as the possibility of implementing nonlinear or non-unitary gates, as well as drawbacks of the proposed model compared to conventional quantum computing are shown.

*J. Adv. Comput. Intell. Intell. Inform.*, Vol.11, No.2, pp. 242-249, 2007.

- [1] D. S. Abrams and S. Lloyd, “Nonlinear Quantum Mechanics Implies Polynomial-Time Solution for
*NP*-Complete and*#P*Problems,” Physical Review Letters, Vol.81, p. 3992, 1998. - [2] L. Adleman, J. DeMarrais, and M. Huang, “Quantum computability,” SIAM Journal on Computing, Vol.26, pp. 1524-1540, 1997.
- [3] N. Bhattacharya, H. B. van Linden van den Heuvell, and R. J. C. Spreeuw, “Implementation of Quantum Search Algorithm using Classical Fourier Optics,” Physical Review Letters, Vol.88, 137901, 2002.
- [4] A. Bolotin, “Quantum mechanical approach to fuzzy logic modeling,” Mathematical and Computer Modelling, Vol.34, No.7-8, pp. 937-945, 2001.
- [5] S. Dasgupta, A. Biswas, and G. S. Agarwal, “Implementing Deutsch-Jozsa algorithm using light shifts and atomic ensembles,” Physical Review A, Vol.7, No.1, 012333, 2005.
- [6] D. Deutsch, “Quantum theory, the church-turing principle and the universal quantum computer,” Proceedings of the Royal Society London, Vol.A400, pp. 97-117, 1985.
- [7] B. D’hooghe, J. Pykacz, and R. R. Zapatrin, “Quantum Computation of Fuzzy Numbers,” International Journal of Theoretical Physics, Vol.43, No.6, pp. 1423-1432, 2004.
- [8] S. Dick, “Toward complex fuzzy logic,” IEEE Transactions on Fuzzy Systems, Vol.13, No.3, pp. 405-414, 2005.
- [9] D. K. Ferry, R. Akis, and J. Harris, “Quantum wave processing,” Superlattices and Microstructures, Vol.30, No.2, pp. 81-94, 2001.
- [10] T. Gopinath and A. Kumar, “Geometric quantum computation using fictitious spin-1/2 subspaces of strongly dipolar coupled nuclear spins,” Physical Review A, Vol.73, 022326, 2006.
- [11] L. K. Grover, “Quantum computers can search arbitrarily large databases by a single query,” Physical Review Letters, Vol.79, pp. 4709-4712, 1997.
- [12] K. Hirota and K. Ozawa, “The concept of fuzzy flip-flop,” IEEE Transactions on Systems, Man and Cybernetics, Vol.19, No.5, pp. 980-997, 1989.
- [13] A. M. Ibrahim, “Bringing fuzzy logic into focus,” Circuits and Devices Magazine, IEEE, Vol.17, No.5, pp. 33-38, 2001.
- [14] M. A. Nielsen and I. L. Chuang, “Quantum Computation and Quantum Information,” Cambridge Univ. Press, Cambridge, U.K., 2000.
- [15] R. Orús and J. I. Latorre, “Universality of entanglement and quantum-computation complexity,” Physical Review A, Vol.69, 052308, 2004.
- [16] S. Ouchi, M. Fujishima, and K. Hoh, “An 8-Qubit Quantum-Circuit Processor,” IEEE Symp. on Circuits and Systems, pp. V-209-212, 2002.
- [17] J. Pykacz, “Fuzzy set ideas in quantum logics,” International Journal of Theoretical Physics, Vol.31, pp. 1767-1783, 1992.
- [18] J. Pykacz, “Fuzzy quantum logics as a basis for quantum probability theory,” International Journal of Theoretical Physics, Vol.37, pp. 281-290, 1998.
- [19] D. Ramot, M. Friedman, G. Langholz, and A. Kandel, “Complex fuzzy logic,” IEEE Transactions on Fuzzy Systems, Vol.11, No.4, pp. 450-461, 2003.
- [20] G. G. Rigatos and S. G. Tzafestas, “Parallelization of a fuzzy control algorithm using quantum computation,” IEEE Transactions on Fuzzy Systems, Vol.10, No.4, pp. 451-460, 2002.
- [21] G. G. Rigatos and S. G. Tzafestas, “Quantum learning for neural associative memories,” Fuzzy Sets and Systems, Vol.157, No.13, pp. 1797-1813, 2006.
- [22] J. Siewert and R. Fazio, “Quantum algorithms for Josephson Networks,” Physical Review Letters, Vol.87, 257905, 2001.
- [23] S. Sinha, T. Munakata, and W. L. Ditto, “Parallel computing with extended dynamical systems,” Physical Review E, Vol.65, 036214, 2002.
- [24] J. Virant, N. Zimic, and M. Mraz, “T-type fuzzy memory cells,” Fuzzy Sets and Systems, Vol.102, No.2, pp. 175-183, 1999.
- [25] H. Peyravi, A. Khoei, and K. Hadidi, “Design of an analog CMOS fuzzy logic controller chip,” Fuzzy Sets and Systems, Vol.132, No.2, pp. 245-260, 2002.

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