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Interference as a computational resource: a tutorial

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Notes

  1. Equivalently, \({\mathbf {N}}{\mathbf {P}}\) is the class of the decision problems that can be verified in polynomial time, when a certificate is given. See Papadimitriou (1994) for details.

  2. In fact, the computation tree in Fig. 5 shows that \(W^2\mid \! 0\rangle =\mid \! 0\rangle\), and it is equally easy to see that \(W^2\mid \! 1\rangle =\mid \! 1\rangle.\)

  3. Characteristic functions \(f_g\) defined as \(f_g(g')=1\) if \(g=g'\) and 0 otherwise clearly form a basis (called the natural basis) for the function space \(G\rightarrow {\mathbb {C}}\).

References

  • Aaronson S (2013) Quantum computing since Democritus. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  • Adleman LM, Demarrais J, Huang M-DA (1997) Quantum computability. SIAM J Comput 26(5):1524–1540

    Article  MathSciNet  MATH  Google Scholar 

  • Ambainis A, Yakaryılmaz A (2017) Automata and quantum computing, a manuscript. arXiv:1507.01988, read 03 Jul

  • Benioff P (1980) The computer as a physical system: a microscopic quantum mechanical Hamiltonian model of computers as represented by turing machines. J Stat Phys 22(5):563–591

    Article  MathSciNet  Google Scholar 

  • Deutsch D (1985) Quantum theory, the Church–Turing principle and the universal quantum computer. Proc R Soc Lond A 400:97–117

    Article  MathSciNet  MATH  Google Scholar 

  • Deutsch D, Jozsa R (1992) Rapid solutions of problems by quantum computation. Proc R Soc Lond A 439:553–558

    Article  MathSciNet  MATH  Google Scholar 

  • Eilenberg S (1974) Automata, languages, and machines. Academic Press, New York

    MATH  Google Scholar 

  • Feynman RP (1982) Simulating physics with computers. Int J Theor Phys 21:467–488

    Article  MathSciNet  Google Scholar 

  • Feynman RP (1987) Negative probability. In: Hiley Basil J, Peat D (eds) Quantum implications: essays in honour of David Bohm. Routledge, Abingdon, pp 235–248

    Google Scholar 

  • Freivalds R (1981) Probabilistic two-way machines. Proc Math Found Comput Sci LNCS 188:33–45

    MathSciNet  MATH  Google Scholar 

  • Gill J (1977) Computational complexity of probabilistic turing machines. SIAM J Comput 6(4):675–695

    Article  MathSciNet  MATH  Google Scholar 

  • Hardy GH, Wright EM (1979) An introduction to the theory of numbers, 5th edn. Oxford University Press, Oxford

    MATH  Google Scholar 

  • Hirvensalo M (2004) Quantum computing, 2nd edn. Springer, Berlin

    Book  MATH  Google Scholar 

  • Hirvensalo M (2008) Various aspects of finite quantum automata. LNCS 5257 (Proceedings of DLT 2008), pp 21–33

  • Hirvensalo M (2010) Quantum automata with open time evolution. Int J Nat Comput Res 1:70–85

    Article  Google Scholar 

  • Hirvensalo M (2012) Mathematics for quantum information processing. In: Rozenberg G, Bck T, Kok J (eds) Handbook of natural computing. Springer, Berlin

    Google Scholar 

  • Manin Y (1980) Computable and uncomputable. Sovetskoye Radio, Moscow (in Russian)

    Google Scholar 

  • Papadimitriou CH (1994) Computational complexity. Pearson, London

    MATH  Google Scholar 

  • Paz A (1971) Introduction to probabilistic automata. Academic Press, Cambridge

    MATH  Google Scholar 

  • Planck M (1900) Annalen der Physik 1:69; Verhandlg dtsch phys Ges 2:202; Verhandlg dtsch phys Ges 2:237; Annalen der Physik 4:553 (1901)

  • Rabin MO (1963) Probabilistic automata. Inf Control 6:230–245

    Article  MATH  Google Scholar 

  • Shor PW (1994) Algorithms for quantum computation: discrete logarithms and factoring. In: Proceedings of the 35th annual symposium on foundations of computer science, 20–22. IEEE Computer Society Press, pp 124–134

  • Shoup V (1997) Lower bounds for discrete logarithms and related problems. Lect Notes Comput Sci 1233:256–266

    Article  MathSciNet  Google Scholar 

  • Simon D (1994) On the power of quantum computation. In: Proceedings of the 35th IEEE symposium on foundations of computer science, pp 116–123

  • Turakainen P (1969a) On probabilistic automata and their generalizations. Annales Academiae Scientiarum Fennicae. Series A 429

  • Turakainen P (1969b) On languages representable in rational probabilistic automata. Annales Academiae Scientiarum Fennicae. Series A 439

  • Sheng Y (1997) Regular languages. In: Rozenberg G, Salomaa A (eds) Handbook of formal languages, vol 1. Springer, Berlin

    Google Scholar 

  • von Neumann J (1932) Mathematische Grundlagen der Quantenmechanik. Springer, Berlin

    MATH  Google Scholar 

  • Young T (1802) The Bakerian lecture: on the theory of light and colours. Philos Trans R Soc Lond 92:12–48

    Article  Google Scholar 

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  1. Department of Mathematics and Statistics, University of Turku, FI-20014, Turku, Finland

    Mika Hirvensalo

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Hirvensalo, M. Interference as a computational resource: a tutorial. Nat Comput 17, 201–219 (2018). https://doi.org/10.1007/s11047-017-9654-x

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