Role of Entanglement in Quantum Neural Networks (QNN)
Affiliation(s)
1 Department of Computer Science, Institute of Engineering & Technology, Khandari Campus, DR. BR Ambedkar University, Agra (U. P.), India. 2 I-11, Gamma-II, Greater Noida (U.P.), India.
1 Department of Computer Science, Institute of Engineering & Technology, Khandari Campus, DR. BR Ambedkar University, Agra (U. P.), India. 2 I-11, Gamma-II, Greater Noida (U.P.), India.
ABSTRACT
Starting with the theoretical basis of quantum computing, entanglement has been explored as one of the key resources required for quantum computation, the functional dependence of the entanglement measures on spin correlation functions has been established and the role of entanglement in implementation of QNN has been emphasized. Necessary and sufficient conditions for the general two-qubit state to be maximally entangled state (MES) have been obtained and a new set of MES constituting a very powerful and reliable eigen basis (different from magic bases) of two-qubit systems has been constructed. In terms of the MES constituting this basis, Bell’s States have been generated and all the qubits of two-qubit system have been obtained. Carrying out the correct computation of XOR function in neural network, it has been shown that QNN requires the proper correlation between the input and output qubits and the presence of appropriate entanglement in the system guarantees this correlation.
Starting with the theoretical basis of quantum computing, entanglement has been explored as one of the key resources required for quantum computation, the functional dependence of the entanglement measures on spin correlation functions has been established and the role of entanglement in implementation of QNN has been emphasized. Necessary and sufficient conditions for the general two-qubit state to be maximally entangled state (MES) have been obtained and a new set of MES constituting a very powerful and reliable eigen basis (different from magic bases) of two-qubit systems has been constructed. In terms of the MES constituting this basis, Bell’s States have been generated and all the qubits of two-qubit system have been obtained. Carrying out the correct computation of XOR function in neural network, it has been shown that QNN requires the proper correlation between the input and output qubits and the presence of appropriate entanglement in the system guarantees this correlation.
KEYWORDS
Entanglement, Maximally Entangled State (MES); Quantum Neural Network (QNN), Eigen Basis, Quantum Associated Memory (Qu AM)
Entanglement, Maximally Entangled State (MES); Quantum Neural Network (QNN), Eigen Basis, Quantum Associated Memory (Qu AM)
Cite this paper
Singh, M. and Rajput, B. (2015) Role of Entanglement in Quantum Neural Networks (QNN). Journal of Modern Physics, 6, 1908-1920. doi: 10.4236/jmp.2015.613196.
Singh, M. and Rajput, B. (2015) Role of Entanglement in Quantum Neural Networks (QNN). Journal of Modern Physics, 6, 1908-1920. doi: 10.4236/jmp.2015.613196.
References
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http://dx.doi.org/10.1007/BF02650179 [2] Shor, P.W. (1994) Algorithms for Quantum Computation: Discrete Logarithm and Factoring. Proceedings of 35th Annual Symposium on Foundations of Computer Science, Santa Fe, 20-22 November 1994, 124-134.
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http://dx.doi.org/10.1023/A:1021695125245 [6] Zak, M. (2000) Quantum Decision-Maker. Information Sciences, 128, 199-215.
http://dx.doi.org/10.1016/s0020-0255(00)00053-0 [7] Bshouty, N.H. and Jackson, J. (1995) Learning DNF over the Uniform Distribution Using a Quantum Example Oracle. Proceedings of the 8th Annual Conference on Computational Learning Theory, Santa Cruz, 5-8 July 1995, 118-127. [8] Li, S.S. and Huang, Y.B. (2008) Entanglement of Superposition of Multistates. International Journal of Quantum Information, 6, 561-565 [9] Li, S.S., Nie, Y.Y., Hong, Z.H., Yi, X.J. and Huang, Y.B. (2008) Controlled Teleportation Using Four-Particle Cluster State. Communications in Theoretical Physics, 50, 633-640 [10] Huang, Y.B., Li, S.S. and Nie, Y.Y. (2009) Controlled Dense Coding between Multi Particles. International Journal of Theoretical Physics, 48, 95-100.
http://dx.doi.org/10.1007/s10773-008-9785-z [11] Li, S.S. (2012) Dense Coding with Cluster State Via Local Measurements. International Journal of Theoretical Physics, 51, 724-730.
http://dx.doi.org/10.1007/s10773-011-0951-3 [12] Wang, Z.S., Wu, C., Feng, X.L., Kwek, L.C., Lai, C.H., Oh, C.H. and Vedral, V. (2007) Nonadiabatic Geometric Quantum Computation. Physical Review A, 76, Article ID: 044303. http://dx.doi.org/10.1103/PhysRevA.76.044303 [13] Wang, Z.S. (2009) Geometric Quantum Computation and Dynamical Invariant Operators. Physical Review A, 79, Article ID: 024304.
http://dx.doi.org/10.1103/PhysRevA.79.024304 [14] Narayanan, A. and Meneer, T. (2000) Quantum Artificial Neural Network Architectures and Components. Information Sciences, 128, 231-255.
http://dx.doi.org/10.1016/S0020-0255(00)00055-4 [15] Behrman, E.C., Nash, L.R., Sleck, J.E., Chandrashekhar, V.G. and Skinner, S.R. (2000) Simulations of Quantum Neural Networks. Information Sciences, 128, 257-269.
http://dx.doi.org/10.1016/s0020-0255(00)00056-6 [16] Ventura, D. and Martinez, T. (2000) Quantum Associative Memory. Information Sciences, 124, 273-296.
http://dx.doi.org/10.1016/S0020-0255(99)00101-2 [17] Ezkov, A., Nifanava, A. and Ventura, D. (2000) Quantum Associative Memory with Distributed Queries. Information Sciences, 128, 271-293.
http://dx.doi.org/10.1016/S0020-0255(00)00057-8 [18] Howell, J., Yeazell, J. and Ventura, D. (2000) Optically Simulating A Quantum Associative Memory. Physical Review A, 62, Article ID: 042303.
http://dx.doi.org/10.1103/PhysRevA.62.042303 [19] Ventura, D. and Martinez, T. (1999) Initializing the Amplitude Distribution of a Quantum State. Foundations of Physics Letters, 12, 547-559.
http://dx.doi.org/10.1023/A:1021695125245 [20] Jennewein, T., Simon, C., Weihs, G., Weinfurter, H. and Zeilinger, A. (2000) Quantum Cryptography with Entangled Photons. Physical Review Letters, 84, 4729-4732.
http://dx.doi.org/10.1103/PhysRevLett.84.4729 [21] Naik, D.S., Peterson, C.G., White, A.G., Burglund, A.J. and Kwiat, P.G. (2000) Entangled State Quantum Cryptography: Eavesdropping on the Ekert Protocol. Physical Review Letters, 84, 4733-4736.
http://dx.doi.org/10.1103/PhysRevLett.84.4733 [22] Tittel, W., Bendel, J., Zbinden, H. and Gisin, N. (2000) Quantum Cryptography Using Entangled Photons in Energy-Time Bell States. Physical Review Letters, 84, 4737-4740.
http://dx.doi.org/10.1103/PhysRevLett.84.4737 [23] Tan, H.T., Zhang, W.M. and Li, G. (2011) Non-Markovian Dynamics of an Open Quantum System with Initial System-Reservoir Correlations: A Nanocavity Coupled to a Coupled-Resonator Optical Waveguide. Physical Review A, 83, Article ID: 032102.
http://dx.doi.org/10.1103/PhysRevA.83.032102 [24] Smirne, A., Breuer, H.P., Piilo, J. and Vacchini, B. (2010) Initial Correlations in Open-Systems Dynamics: The Jaynes-Cummings Model. Physical Review A, 84, Article ID: 062114.
http://dx.doi.org/10.1103/PhysRevA.82.062114 [25] Benenti, G. and Casati, G. (2009) How Complex Is Quantum Motion? Physical Review E, 79, Article ID: 025201.
http://dx.doi.org/10.1103/PhysRevE.79.025201 [26] Hill, S. and Wooters, W.K. (1997) Entanglement of a Pair of Quantum Bits. Physical Review Letters, 78, 5022-5025.
http://dx.doi.org/10.1103/PhysRevLett.78.5022 [27] Wooters, W.K. (1998) Entanglement of Formation of an Arbitrary State of Two Qubits. Physical Review Letters, 80, 2245-2248.
http://dx.doi.org/10.1103/PhysRevLett.80.2245 [28] Monvol, C., Meekhof, D.M., King, B.E., Itano, W.M. and Wineland, D.J. (1995) Demonstration of a Fundamental Quantum Logic Gat. Physical Review Letters, 75, 4714-4718.
http://dx.doi.org/10.1103/PhysRevLett.75.4714
[1] Feynman, R.P. (1982) Simulating Physics with Computers. International Journal of Theoretical Physics, 21, 467-488.
http://dx.doi.org/10.1007/BF02650179 [2] Shor, P.W. (1994) Algorithms for Quantum Computation: Discrete Logarithm and Factoring. Proceedings of 35th Annual Symposium on Foundations of Computer Science, Santa Fe, 20-22 November 1994, 124-134.
http://dx.doi.org/10.1109/sfcs.1994.365700 [3] Grover, L.K. (1996) A Fast Quantum Mechanical Algorithm for Data Base Search. Proceedings of 28th Annual ACM Symposium on Theory of Computing, Philadelphia, 22-24 May 1996, 212-219. [4] Simon, D. (1997) On the power of Quantum Computation. SIAM Journal on Computing, 26, 1474-1483. [5] Ventura, D. and Martinez, T. (1999) Initializing Amplitude Distribution of a Quantum State. Foundations of Physics Letters, 12, 547-559.
http://dx.doi.org/10.1023/A:1021695125245 [6] Zak, M. (2000) Quantum Decision-Maker. Information Sciences, 128, 199-215.
http://dx.doi.org/10.1016/s0020-0255(00)00053-0 [7] Bshouty, N.H. and Jackson, J. (1995) Learning DNF over the Uniform Distribution Using a Quantum Example Oracle. Proceedings of the 8th Annual Conference on Computational Learning Theory, Santa Cruz, 5-8 July 1995, 118-127. [8] Li, S.S. and Huang, Y.B. (2008) Entanglement of Superposition of Multistates. International Journal of Quantum Information, 6, 561-565 [9] Li, S.S., Nie, Y.Y., Hong, Z.H., Yi, X.J. and Huang, Y.B. (2008) Controlled Teleportation Using Four-Particle Cluster State. Communications in Theoretical Physics, 50, 633-640 [10] Huang, Y.B., Li, S.S. and Nie, Y.Y. (2009) Controlled Dense Coding between Multi Particles. International Journal of Theoretical Physics, 48, 95-100.
http://dx.doi.org/10.1007/s10773-008-9785-z [11] Li, S.S. (2012) Dense Coding with Cluster State Via Local Measurements. International Journal of Theoretical Physics, 51, 724-730.
http://dx.doi.org/10.1007/s10773-011-0951-3 [12] Wang, Z.S., Wu, C., Feng, X.L., Kwek, L.C., Lai, C.H., Oh, C.H. and Vedral, V. (2007) Nonadiabatic Geometric Quantum Computation. Physical Review A, 76, Article ID: 044303. http://dx.doi.org/10.1103/PhysRevA.76.044303 [13] Wang, Z.S. (2009) Geometric Quantum Computation and Dynamical Invariant Operators. Physical Review A, 79, Article ID: 024304.
http://dx.doi.org/10.1103/PhysRevA.79.024304 [14] Narayanan, A. and Meneer, T. (2000) Quantum Artificial Neural Network Architectures and Components. Information Sciences, 128, 231-255.
http://dx.doi.org/10.1016/S0020-0255(00)00055-4 [15] Behrman, E.C., Nash, L.R., Sleck, J.E., Chandrashekhar, V.G. and Skinner, S.R. (2000) Simulations of Quantum Neural Networks. Information Sciences, 128, 257-269.
http://dx.doi.org/10.1016/s0020-0255(00)00056-6 [16] Ventura, D. and Martinez, T. (2000) Quantum Associative Memory. Information Sciences, 124, 273-296.
http://dx.doi.org/10.1016/S0020-0255(99)00101-2 [17] Ezkov, A., Nifanava, A. and Ventura, D. (2000) Quantum Associative Memory with Distributed Queries. Information Sciences, 128, 271-293.
http://dx.doi.org/10.1016/S0020-0255(00)00057-8 [18] Howell, J., Yeazell, J. and Ventura, D. (2000) Optically Simulating A Quantum Associative Memory. Physical Review A, 62, Article ID: 042303.
http://dx.doi.org/10.1103/PhysRevA.62.042303 [19] Ventura, D. and Martinez, T. (1999) Initializing the Amplitude Distribution of a Quantum State. Foundations of Physics Letters, 12, 547-559.
http://dx.doi.org/10.1023/A:1021695125245 [20] Jennewein, T., Simon, C., Weihs, G., Weinfurter, H. and Zeilinger, A. (2000) Quantum Cryptography with Entangled Photons. Physical Review Letters, 84, 4729-4732.
http://dx.doi.org/10.1103/PhysRevLett.84.4729 [21] Naik, D.S., Peterson, C.G., White, A.G., Burglund, A.J. and Kwiat, P.G. (2000) Entangled State Quantum Cryptography: Eavesdropping on the Ekert Protocol. Physical Review Letters, 84, 4733-4736.
http://dx.doi.org/10.1103/PhysRevLett.84.4733 [22] Tittel, W., Bendel, J., Zbinden, H. and Gisin, N. (2000) Quantum Cryptography Using Entangled Photons in Energy-Time Bell States. Physical Review Letters, 84, 4737-4740.
http://dx.doi.org/10.1103/PhysRevLett.84.4737 [23] Tan, H.T., Zhang, W.M. and Li, G. (2011) Non-Markovian Dynamics of an Open Quantum System with Initial System-Reservoir Correlations: A Nanocavity Coupled to a Coupled-Resonator Optical Waveguide. Physical Review A, 83, Article ID: 032102.
http://dx.doi.org/10.1103/PhysRevA.83.032102 [24] Smirne, A., Breuer, H.P., Piilo, J. and Vacchini, B. (2010) Initial Correlations in Open-Systems Dynamics: The Jaynes-Cummings Model. Physical Review A, 84, Article ID: 062114.
http://dx.doi.org/10.1103/PhysRevA.82.062114 [25] Benenti, G. and Casati, G. (2009) How Complex Is Quantum Motion? Physical Review E, 79, Article ID: 025201.
http://dx.doi.org/10.1103/PhysRevE.79.025201 [26] Hill, S. and Wooters, W.K. (1997) Entanglement of a Pair of Quantum Bits. Physical Review Letters, 78, 5022-5025.
http://dx.doi.org/10.1103/PhysRevLett.78.5022 [27] Wooters, W.K. (1998) Entanglement of Formation of an Arbitrary State of Two Qubits. Physical Review Letters, 80, 2245-2248.
http://dx.doi.org/10.1103/PhysRevLett.80.2245 [28] Monvol, C., Meekhof, D.M., King, B.E., Itano, W.M. and Wineland, D.J. (1995) Demonstration of a Fundamental Quantum Logic Gat. Physical Review Letters, 75, 4714-4718.
http://dx.doi.org/10.1103/PhysRevLett.75.4714