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Quantum and Biocomputing – Common Notions and Targets

  • Mika Hirvensalo

Abstract

Biocomputing and quantum computing are both relatively novel areas of information processing sciences under the umbrella natural computing established in the late twentieth century. From the practical point of view one can say that in both bio and quantum paradigms, the purpose is to replace the traditional media of computing by an alternative. Biocomputing is based on an appropriate treatment of biomolecules, and quantum computing is based on the physical realization of computation on systems so small that they must be described by using quantum mechanics. The efficiency of the proposed biomolecular computing is based on massive parallelism, which is implementable by already existing technology for small instances. In a sense, also quantum computing involves parallelism. From time to time, there are proposals or attempts to create a uniform approach to both biocomputational and quantum parallelism. The main purpose of this article is the explain why this a very challenging task. For this aim, we present the usual mathematical formalism needed to speak about quantum computing and compare quantum parallelism to its biomolecular counterpart.

Keywords

Quantum Computing Quantum Computer Quantum Algorithm Quantum Circuit Bell Inequality 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Abbreviations

DNA

deoxyribonucleic acid

EPR

Einstein–Podolsky–Rosen

NP

neuritic plaque

RSA

Rivest, Shamir, Adleman

References

  1. 59.1.
    L.M. Adleman: Molecular computation of solutions to combinatorial problems, Science 266(11), 1021–1024 (1994)CrossRefGoogle Scholar
  2. 59.2.
    R.J. Lipton: DNA solution of hard computational problems, Science 268(5210), 542–545 (1995)CrossRefGoogle Scholar
  3. 59.3.
    M.L. Minsky, S.A. Papert: Perceptrons (MIT Press, Cambridge 1969)zbMATHGoogle Scholar
  4. 59.4.
    P.W. Shor: Algorithms for quantum computation: Discrete log and factoring, Proc. 35th Annu. IEEE Symp. Found. Comput. Sci. (1994) pp. 20–22Google Scholar
  5. 59.5.
    A.M. Turing: On computable numbers, with an application to the entscheidungsproblem, Proc. Lond. Math. Soc. 2(42), 230–265 (1936)MathSciNetzbMATHGoogle Scholar
  6. 59.6.
    J. von Neumann: Thermodynamik quantummechanischer Gesamheiten, Nachr. Ges. Wiss. Gött. 1, 273–291 (1927)Google Scholar
  7. 59.7.
    J. von Neumann: Mathematische Grundlagen der Quantenmechanik (Springer, Berlin 1932)zbMATHGoogle Scholar
  8. 59.8.
    P.A. Benioff: 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 (1980)MathSciNetCrossRefGoogle Scholar
  9. 59.9.
    P.A. Benioff: Quantum mechanical Hamiltonian models of discrete processes that erase their own histories: Application to turing machines, Int. J. Theor. Phys. 21(3/4), 177–202 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 59.10.
    R.P. Feynman: Simulating physics with computers, Int. J. Theor. Phys. 21(6/7), 467–488 (1982)MathSciNetCrossRefGoogle Scholar
  11. 59.11.
    D. Deutsch: Quantum theory, the Church–Turing principle and the universal quantum computer, Proc. R. Soc. A 400, 97–117 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 59.12.
    D. Deutsch: Quantum computational networks, Proc. R. Soc. A 425, 73–90 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 59.13.
    D. Deutsch, R. Jozsa: Rapid solutions of problems by quantum computation, Proc. R. Soc. A 439, 553 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 59.14.
    D.R. Simon: On the power of quantum computation, Proc. 35th Annu. IEEE Symp. Found. Comput. Sci. (1994) pp. 116–123Google Scholar
  15. 59.15.
    E. Bernstein, U. Vazirani: Quantum complexity theory, SIAM J. Comput. 26(5), 1411–1473 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 59.16.
    Clay Mathematics Institute: http://www.claymath.org/millennium/P_vs_NP/
  17. 59.17.
    M. Hirvensalo: Quantum Computing, 2nd edn. (Springer, Berlin, Heidelberg 2004)CrossRefzbMATHGoogle Scholar
  18. 59.18.
    M. Hirvensalo: Mathematics for quantum information processing. In: Handbook of Natural Computing, ed. by G. Rozenberg, T. Bäck, J. Kok (Springer, Berlin, Heidelberg 2011)Google Scholar
  19. 59.19.
    M.A. Nielsen, I.L. Chuang: Quantum Computation and Quantum Information (Cambridge Univ. Press, Cambridge 2000)zbMATHGoogle Scholar
  20. 59.20.
    M. Hirvensalo: EPR Paradox and Bell Inequalities, Bulletin EATCS 92, 115–139 (2007)MathSciNetzbMATHGoogle Scholar
  21. 59.21.
    D. Bohm: Quantum Theory (Prentice-Hall, Englewood Cliffs 1951) pp. 614–619Google Scholar
  22. 59.22.
    A. Einstein, B. Podolsky, N. Rosen: Can quantum-mechanical description of physical reality be considered complete?, Phys. Rev. 47, 777–780 (1935)CrossRefzbMATHGoogle Scholar
  23. 59.23.
    J.S. Bell: On the Einstein–Podolsky–Rosen paradox, Physics 1, 195–200 (1964)Google Scholar
  24. 59.24.
    R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J.G. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, A. Zeilinger: Free-space distribution of entanglement and single photons over 144 km, Nat. Phys. 3(7), 481–486 (2007)CrossRefGoogle Scholar
  25. 59.25.
    A. Barenco, C.H. Bennett, R. Cleve, D.P DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. Smolin, H. Weinfurter: Elementary gates for quantum computation, Phys. Rev. A 52(5), 3457–3467 (1995)CrossRefGoogle Scholar
  26. 59.26.
    A. Kondacs, J. Watrous: On the power of quantum finite state automata, Proc. 38th Annu. Symp. Found. Comput. Sci. (1997) pp. 66–75Google Scholar
  27. 59.27.
    C. Moore, J.P. Crutchfield: Quantum automata and quantum grammars, Theor. Comput. Sci. 237(1/2), 275–306 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 59.28.
    C.H. Bennett: Logical reversibility of computation, IBM J. Res. Dev. 17, 525–532 (1973)CrossRefMathSciNetzbMATHGoogle Scholar
  29. 59.29.
    M. Hirvensalo: Quantum automata with open time evolution, Int. J. Nat. Comput. Res. 1, 70–85 (2010)CrossRefGoogle Scholar
  30. 59.30.
    L.K. Grover: A fast quantum-mechanical algorithm for database search, Proc. 28th Annu. ACM Symp. Theory Comput. (1996) pp. 212–219Google Scholar
  31. 59.31.
    J. Kempe: Discrete quantum walks hit exponentially faster, Probab. Theory Relat. Fields 133(2), 215–235 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 59.32.
    J. Kempe: Quantum random walks – an introductory overview, Contemp. Phys. 44(4), 307–327 (2003)MathSciNetCrossRefGoogle Scholar
  33. 59.33.
    D. Aharonov, W. van Dam, J. Kempe, Z. Landau, S. Lloyd, O. Regev: Adiabatic quantum computation is equivalent to standard quantum computation, SIAM J. Comput. 37, 166–194 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 59.34.
    C.H. Papadimitriou: Computational Complexity (Addison-Wesley, Reading 1994)zbMATHGoogle Scholar
  35. 59.35.
    R. Beals, H. Buhrman, R. Cleve, M. Mosca, R. Wolf: Quantum lower bounds by polynomials, Journal ACM 48(4), 778–797 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  36. 59.36.
    A. Ambainis: Quantum lower bounds by quantum arguments, J. Comput. System Sci. 64(4), 750–767 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 59.37.
    J.I. Cirac, P. Zoller: Quantum computations with cold trapped ions, Phys. Rev. Lett. 74, 4091–4094 (1995)CrossRefGoogle Scholar
  38. 59.38.
    I.L. Chuang, N. Gershenfeld, M. Kubinec: Experimental implementation of fast quantum searching, Phys. Rev. Lett. 80, 3408–3411 (1998)CrossRefGoogle Scholar
  39. 59.39.
    J.L. OʼBrien: Optical quantum computing, Science 318, 1567–1570 (2007)CrossRefGoogle Scholar
  40. 59.40.
    C. Negrevergne, T.S. Mahesh, C.A. Ryan, M. Ditty, F.-Y. Cyr-Racine, W. Power, N. Boulant, T. Havel, D.G. Cory, R. Laflamme: Benchmarking quantum control methods on a 12-qubit system, Phys. Rev. Lett. 96, 170501 (2006)CrossRefGoogle Scholar
  41. 59.41.
    S. Beauregard: Circuit for Shorʼs algorithm using 2n + 3 qubits, Quantum Inform. Comput. 3(2), 175–185 (2003)MathSciNetzbMATHGoogle Scholar
  42. 59.42.
    V.V. Nelayev, K.N. Dovzhik, V.V. Lyskouski: Quantum effects in biomolecular structures, Rev. Adv. Mater. Sci. 20, 42–47 (2009)Google Scholar

Copyright information

© Springer-Verlag 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TurkuTurkuFinland

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