Abstract
A quantum computer is usually modeled mathematically as a Quantum Turing Machine (QTM) or a uniform family of quantum circuits, which is equivalent to a quantum Turing machine. QTM is a quantum version of the classical Turing machine described in Chap. 2. QTM was introduced by Deutsch and has been extensively studied by Bernstein and Vasirani. The basic properties of the quantum Turing machine and quantum circuits will be described in this chapter. In the last section of the present chapter, we introduce a generalized QTM.
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Barenco, A., Bennett, C.H., Cleve, R., DiVicenzo, D., Margolus, N., Shor, P., Sleator, T., Smolin, J., Weinfurter, H.: Elementary gates for quantum computation. Phys. Rev. A 52, 3457–3467 (1995)
Bernstein, E., Vazirani, U.: Quantum complexity theory. In: Proc. of the 25th Annual ACM Symposium on Theory of Computing, pp. 11–22. ACM, New York (1993). SIAM J. Comput. 26, 1411 (1997)
Deutsch, D.: Quantum theory, the Church-Turing principle and the universal quantum computer. Proc. R. Soc. Lond., Ser. A 400, 96–117 (1985)
Deutsch, D.: Quantum computational networks. Proc. R. Soc. Lond. Ser. A 425, 73–90 (1989)
Freudenberg, W., Ohya, M., Watanabe, N.: Generalized Fock space approach to Fredkin-Toffoli-Milburn gate. TUS preprint (2001)
Grover, L.K.: Quantum mechanics helps in searching for a needle in a haystack. Phys. Rev. Lett. 78(2), 325 (1997)
Iriyama, S., Ohya, M., Volovich, I.V.: Generalized quantum Turing machine and its application to the SAT chaos algorithm. In: QP-PQ: Quantum Prob. White Noise Anal., Quantum Information and Computing, vol. 19, pp 204–225. World Scientific, Singapore (2006)
Iriyama, S., Miyadera, T., Ohya, M.: Note on a universal quantum Turing machine. Phys. Lett. A 372, 5120–5122 (2008)
Iriyama, S., Ohya, M.: Review on quantum chaos algorithm and generalized quantum Turing machine. In: QP-PQ: Quantum Prob. White Noise Anal., Quantum Bio-Informatics, vol. 21, pp. 126–141. World Scientific, Singapore (2008)
Miyadera, T., Ohya, M.: On halting process of quantum Turing machine. Open Syst. Inf. Dyn. 12(3), 261–264 (2006)
Myers, J.M.: Can a universal quantum computer be fully quantum? Phys. Rev. Lett. 78, 1823 (1997)
Ohya, M., Masuda, N.: NP problem in quantum algorithm. Open Syst. Inf. Dyn. 7(1), 33–39 (2000)
Ohya, M., Volovich, I.V.: Quantum computing, NP-complete problems and chaotic dynamics. In: Hida, T., Saito, K. (eds.) Quantum Information, pp. 161–171. World Scientific, Singapore (2000)
Ohya, M., Volovich, I.V.: Quantum computing and chaotic amplification. J. Opt. B 5(6), 639–642 (2003)
Ohya, M., Volovich, I.V.: New quantum algorithm for studying NP-complete problems. Rep. Math. Phys. 52(1), 25–33 (2003)
Ozawa, M.: Quantum nondemolition monitoring of universal quantum computers. Theor. Inf. Appl. 34, 379 (2000)
Shi, Y.: Remarks on universal quantum computer. Phys. Lett. A 293, 277 (2002)
Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26, 1484 (1997)
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Ohya, M., Volovich, I. (2011). Mathematical Models of Quantum Computer. In: Mathematical Foundations of Quantum Information and Computation and Its Applications to Nano- and Bio-systems. Theoretical and Mathematical Physics. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0171-7_11
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