Skip to main content

Advertisement

SpringerLink
Log in

A review of procedures to evolve quantum algorithms

  • Original Paper
  • Published:
Genetic Programming and Evolvable Machines Aims and scope Submit manuscript

Abstract

There exist quantum algorithms that are more efficient than their classical counterparts; such algorithms were invented by Shor in 1994 and then Grover in 1996. A lack of invention since Grover’s algorithm has been commonly attributed to the non-intuitive nature of quantum algorithms to the classically trained person. Thus, the idea of using computers to automatically generate quantum algorithms based on an evolutionary model emerged. A limitation of this approach is that quantum computers do not yet exist and quantum simulation on a classical machine has an exponential order overhead. Nevertheless, early research into evolving quantum algorithms has shown promise. This paper provides an introduction into quantum and evolutionary algorithms for the computer scientist not familiar with these fields. The exciting field of using evolutionary algorithms to evolve quantum algorithms is then reviewed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Notes

  1. A more detailed and structured account of Feynman’s idea was given in his book [32].

  2. An oracle problem is such that some properties of a function are desired to be discovered, whereby that function is given as a black box. The code of a black box function is unknown, but the function’s results for given inputs can be calculated.

  3. A complete quantum solution with a probability of one was presented by Cleve, Ekert, Macchiavello and Mosca in 1998 [20].

  4. Simon’s algorithm, like Deutsch’s, lacks practical application.

  5. Chapter 16 of Hardy and Steeb’s book [48] contains a good introduction to Hilbert Space.

  6. One of many news articles about Orion can be viewed at http://arstechnica.com/articles/paedia/hardware/quantum.ar.

  7. To be orthogonal means to be linearly independent: a precise treatment of the orthogonality condition is given by MathWorld [available at http://mathworld.wolfram.com/OrthogonalityCondition.html.

  8. A known state of superposition is where all the amplitudes are known.

  9. The term quantum effects is used as an umbrella term to encompass effects such as superposition and entanglement that exist in quantum computing, but not classical computing.

  10. Shor’s algorithm is not proof of an exponential speed gap as the classical complexity of factoring is not known definitively.

  11. By convention a qubit begins in a basis state, usually assumed to be \(|{0}\rangle\).

  12. There is some inconsistency in the literature about the maximum number of qubits that a gate should act upon; for example, Shor [96] states the maximum should be two qubits while Spector et al. [102] state that the maximum is a few qubits.

  13. Variations in the quantum circuit notation of different gates exist, but they are only minor and should be understood by a reader who has read this paper.

  14. Due to the unitary restriction and linearity, transformations are fully specified by their effect on the basis states [90].

  15. This explanation of the quantum teleportation algorithm is based on the explanation given by Landry [61].

  16. Quantum teleportation of one qubit has been realised experimentally [16].

  17. The exact probability of observing a solution depends upon the number of solutions in the search space.

  18. Mathematica is a comprehensive mathematical software package, details are available on the their website: http://www.wolfram.com/products/mathematica/index.htm.

  19. Multimodal problems are problems that arise in cases with a large number of locally optimum solutions.

  20. An example of overlap is the GA research of Surry and Radcliffe [112], which overlaps into the ES field.

  21. Note that with GAs and GPAs Darwinian selection occurs before genetic modification, while with ES and EP this order is commonly reversed [23].

  22. The desired level of optimisation is set as a parameter in terms of the fitness function.

  23. The return value of a quantum gate node is a valid index of a qubit it acted upon. For example, the CNOT gate is usually defined to return the index of the control qubit.

  24. There is also a trivial mapping of a quantum circuit to a tree-structured program, based on establishing a sequence of gates, that is further described in Leier’s thesis [64].

  25. Only discrete parameters were allowed.

  26. [] indicates optionality.

  27. It is technically not a GP model due to the fixed length representation of individuals.

  28. Here Spector et al. are using the term correct to mean correct greater than or equal to 52% of the time.

  29. Two maximally entangled qubits is in fact the first Bell state as described in Sect. 2.2.5.

  30. Rubinstein [91] also included an observe gate capable of partial measurement as discussed in Sect. 4.3.4.

  31. Note that the result of the linear-tree models applied to 1-SAT is also contained in a paper by Leier and Banzhaf [65].

  32. A formal definition of the hidden subgroup problem can be found at http://en.wikipedia.org/wiki/Hidden_subgroup_proble.

  33. These publications have resulted from research presented in Massey’s PhD thesis [79].

  34. Wall’s GP C++ library is available from http://lancet.mit.edu/ga.

  35. Massey et al. [78] specify the exact gate set used by each software version.

  36. Massey et al. [78] specify the exact differences between the software versions.

  37. The importance of QFTs (described well in the book [85]) is based on the fact it is the major building block of Shor’s famous algorithm [94].

  38. The population size was reduced to 50 after the second generation.

  39. The fitness function included a component that penalised individuals without the ‘correct’ known number of Swap gates.

  40. Within the scope of this paper, Phase and π/8 gates can be thought of as reflection (about a basis state) and rotation gates respectively.

  41. Clark and Stepney [19] suggest an approach to search whereby classical search is first used to reduce the search space so that Grover’s algorithm can then be used.

References

  1. T. Bäck, Evolutionary Algorithms in Theory and Practice: Evolution Strategies, Evolutionary Programming, Genetic Algorithms (Oxford Univeristy Press, 1996)

  2. T. Bäck, D.B. Fogel, Z. Michalewicz (ed.), Evolutionary Computation 1: Basic Algorithms and Operators (Institute of Physics, 2000)

  3. T. Bäck, D.B. Fogel, Z. Michalewicz (ed.), Evolutionary Computation 2: Advanced Algorithms and Operators (Institute of Physics, 2000)

  4. W. Banzhaf, P. Nordin, R. Keller, F. Francone, Genetic Programming—An Introduction (dpunkt Heidelberg and Morgan Kaufmann Publishers, San Francisco, 1998)

  5. A. Barenco, A universal two-bit gate for quantum computation, in Proceedings of the Royal Society of London A, vol. 449 (1995), pp. 679–683

  6. 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)

    Article  Google Scholar 

  7. H. Barnum, H.J. Bernstein, L. Spector, A quantum circuit for OR. ArXiv Quantum Physics e-prints quant-ph/9907056v3 (October 1999)

  8. H. Barnum, H.J. Bernstein, L. Spector, Quantum circuits for OR and AND of ORs. J. Phys. A Math. Gen. 33(45), 8047–8057 (November 2000)

  9. R.K. Belew, M.D. Vose (eds.), Foundations of Genetic Algorithms, vol. 4 (Morgan Kaufmann, 1997)

  10. P. 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)

    Article  MathSciNet  Google Scholar 

  11. C.H. Bennett, Logical reversibility of computation. IBM J. Res. Dev. 17, 525–532 (1973)

    Article  MATH  Google Scholar 

  12. C.H. Bennett, E. Bernstein, G. Brassard, U.V. Vazirani, Strengths and weaknesses of quantum computing. Soc. Ind. Appl. Math. J. Comput. 26, 1510–1523 (1994)

    MathSciNet  Google Scholar 

  13. E. Bernstein, U.V. Vazirani, Quantum complexity theory, in Proceedings of the 25th Annual ACM Symposium on Theory of Computation (1993), pp. 11–20

  14. T. Blickle, Tournament selection, in Evolutionary Computation 1: Basic Algorithms and Operators, ed. by T. Bäck, D.B. Fogel, Z. Michalewicz (Institute of Physics, 2000), pp. 181–186

  15. S. Bornholdt, Genetic algorithms, in Non-Standard Computation, ed. by T. Gramß, S. Bornholdt, M. Groß, M. Mitchell, T. Pellizzari (WILEY-VCH, Weinheim, Germany, 1998), pp. 141–178

    Google Scholar 

  16. D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibi, H. Weinfurther, A. Zeilinger, Experimental quantum teleportation. Nature 390, 575–579 (1997)

    Article  Google Scholar 

  17. S.L. Braunstein, Quantum computation: a tutorial. Available from author’s homepage at the Department of Computer Science, York University: http://www-users.cs.york.ac.uk/~schmuel/, 1995

  18. C.S. Calude, De-quantizing the solution of Deutsch’s problem. Int. J. Quantum Inf. 5(3), 409–415 (June 2007). Preprint available at: http://arxiv.org/abs/quant-ph/0610220

  19. J.A. Clark, S. Stepney, Fusing natural computational paradigms for cryptanalysis. Or, using heuristic search to bring cryptanalysis problems within quantum computational range, in Proceedings of the 2006 Congress on Evolutionary Computation (CEC2006) (2006), pp. 200–206

  20. R. Cleve, A. Ekert, C. Macchiavello, M. Mosca, Quantum algorithms revisited, in Proceedings of the Royal Society of London A, vol. 454 (1998), pp. 339–354

  21. N.L. Cramer A representation for the adaptive generation of simple sequential programs, in Proceedings of an International Conference on Genetic Algorithms and their Applications, ed. by J.J. Grefenstette (July 1985), pp. 183–187

  22. C. Darwin, On the Origin of Species by Means of Natural Selection or the Preservation of Favoured Races in the Struggle for Life (Murray, London, 1859)

  23. K. Deb, Introduction into selection, in Evolutionary Computation 1: Basic Algorithms and Operators, ed. by T. Bäck, D.B. Fogel, Z. Michalewicz (Institute of Physics, 2000), pp. 166–171

  24. D. Deutsch, Quantum theory, the church-turing principle and the universal quantum computer 400, in Proceedings of the Royal Society of London A (1985), pp. 97–117

  25. D. Deutsch, Quantum computational networks, in Proceedings of the Royal Society of London A 425 (1989), pp. 73–90

  26. D. Deutsch, R. Jozsa Rapid solution of problems by quantum computation, in Proceedings of the Royal Society of London Series A, vol. A439 (1992), pp. 553–558

  27. D. Dickmanns, J. Schmidhuber, A. Winklhofer, Der genetische algorithmus: Eine implementierung in prolog. Tech. rep. (Institut für Informatik, Technische Universität Mänchen, 1987)

  28. S. Ding, Z. Jin, Q. Yang, Evolving quantum oracles with hybrid quantum-inspired evolutionary algorithm. ArXiv Quantum Physics e-prints quant-ph/0610105 (October 2006)

  29. P. Dirac, The Principles of Quantum Mechanics, 4th edn. (Oxford University Press, 1958)

  30. R. Feynman, Simulating physics with computers. Int. J. Theor. Phys. 21, 467–488 (1982)

    Google Scholar 

  31. R.P. Feynman, There’s plenty of room at the bottom: an invitation to enter a new field of physics. Speech at the Annual Meeting of the American Physical Society, December 1959. It is available online at http://www.zyvex.com/nanotech/feynman.html

  32. R.P. Feynman, D.H. Gilbert DH (eds.), Miniaturization (Reinhold, New York, 1961), pp. 282–295

  33. D.B. Fogel, Phenotypes, genotypes, and operators in evolutionary computation, in Computational Intelligence: Theory and Applications (5th Fuzzy Days Berlin, Springer-Verlag, 1995), pp. 337–342

  34. G.D. Forney Jr., S. Guha, Simple rate-1/3 convolutional and tail-biting quantum error-correcting codes, in Proceedings of the IEEE International Symposium on Information Theory (September 2005), pp. 1028–1032

  35. L. Fortnow, J. Rogers, Complexity limitations on quantum computation. J. Comput. Syst. Sci. 59(2), 240–252 (1999) (Special issue for selected papers from the 13th IEEE Conference on Computational Complexity)

    Google Scholar 

  36. A.A. Freitas, Data Mining and Knowledge Discovery with Evolutionary Algorithms (Springer-Verlag, 2002)

  37. A. Galindo, M. Martín-Delgado, Information and computation: classical and quantum aspects. Rev Modern Phys. 74(2), 347–423 (May 2002). Preprint available at: http://arxiv.org/abs/quant-ph/0112105

  38. G.A. Giraldi, R. Portugal, R.N. Thess, Genetic algorithms and quantum computation. ArXiv Quantum Physics e-prints quant-ph/0610105 (March 2004)

  39. D.E. Goldberg, Genetic Algorithms in Search, Optimisation, and Machine Learning (Addison Wesley, 1989)

  40. T. Gramß (1998) The theory of quantum computation: an introduction, in Non-Standard Computation ed. by T. Gramß, S. Bornholdt, M. Groß, M. Mitchell, T. Pellizzari (WILEY-VCH, Weinheim, Germany), pp. 141–178

  41. J. Grefenstette, Proportional selection and sampling algorithms, in Evolutionary Computation 1: Basic Algorithms and Operators, ed. by T. Bäck, D.B. Fogel, Z. Michalewicz (Institute of Physics, 2000), pp. 172–180

  42. J. Grefenstette, Rank-based selection, in Evolutionary Computation 1: Basic Algorithms and Operators, ed. by T. Bäck, D.B. Fogel, Z. Michalewicz (Institute of Physics, 2000), pp. 187–194

  43. M. Groß, Molecular computing, in Non-Standard Computation, ed. by T. Gramß, S. Bornholdt, M. Groß, M. Mitchell, T. Pellizzari (WILEY-VCH, Weinheim, Germany, 1998), pp. 15–58

    Chapter  Google Scholar 

  44. L.K. Grover, A fast quantum mechanical algorithm for database search, in Proceedings of the 18th Annual ACM Symposium on the History of Computing (Philadelphia, Pennsylvania, May 1996), pp. 212–219

  45. S. Gudder, Quantum automata: an overview. Int. J. Theor. Phys. 28(9), 2261–2282 (1999)

    Article  MathSciNet  Google Scholar 

  46. H. Häffner, W. Hänsel, C.F. Roos, J. Benhelm, D. Chek-al-kar, M. Chwalla, T. Körber, U.D. Rapol, M. Riebe, P.O. Schmidt, C. Becher, O. Gühne, W. Dür, R. Blatt, Scalable multiparticle entanglement of trapped ions. Nature 438(1), 643–646 (December 2005)

    Google Scholar 

  47. Y. Hardy, W.-H. Steeb, Entangled quantum states and a C++ implementation. Int. J. Modern Phys. C 11, 69–77 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  48. Y. Hardy, W.-H. Steeb, Classical and Quantum Computing, with C++ and Java Simulations (Birkhäuser Verlag Berlin, Germany, 2001)

    MATH  Google Scholar 

  49. T. Hogg, Solving highly constrained search problems with quantum computers. J. Artif. Intell. Res. 10, 39–66 (1999)

    MATH  MathSciNet  Google Scholar 

  50. J.H. Holland, Adaptation in Natural and Artificial Systems (The MIT Press, Cambridge, 1975)

    Google Scholar 

  51. R. Huber, T. Schell, Mixed size tournament selection. Soft Comput. Fus. Found. Methodol. Appl. 6(6), 449–455 (September 2002)

  52. T.A. Hungerford, Algebra (Springer-Verlag, New York, 1974)

    MATH  Google Scholar 

  53. B.A. Julstrom It’s all the same to me: Revisiting rank-based probabilities and tournaments, in Proceedings of the Congress on Evolutionary Computation, vol. 2 (IEEE Press, Piscataway, New Jersey, 1999), pp. 1501–1505

  54. W. Kantschik, W. Banzhaf, Linear-Tree GP and its comparison with other GP structures, in Genetic Programming, Proceedings of EuroGP’2001, ed. by J.F. Miller, M. Tomassini, P.L. Lanzi, C. Ryan, A.G.B. Tettamanzi, W.B. Langdon. Lecture Notes in Computer Science, vol. 2038 (Springer-Verlag, 2001), pp. 302–312

  55. W. Kantschik, W. Banzhaf, Linear-graph GP—a new GP structure, in Genetic Programming, Proceedings of EuroGP’2002, ed. by J.A. Foster, E. Lutton, J. Miller, C. Ryan, A.G.B. Tettamanzi. Lecture Notes in Computer Science, vol. 2278 (Springer-Verlag, 2002), pp. 83–92

  56. M.H.A. Khan, M. Perkowski, Genetic algorithm based synthesis of multi-output ternary functions using quantum cascade of generalized ternary gates, in Proceedings of 2004 Congress on Evolutionary Computation (June 2004)

  57. J.R. Koza, Genetic Programming: On the Programming of Computers by Means of Natural Selection (The MIT Press, Cambridge 1992)

    MATH  Google Scholar 

  58. J.R. Koza, Genetic Programming II: Automatic Discovery of Reusable Programs (The MIT Press, Cambridge, 1994)

    MATH  Google Scholar 

  59. J.R. Koza, F.H. Bennett III, D. Andre, M.A. Keane, Genetic Programming III: Darwinian Invention and Problem Solving. (Morgan Kaufmann, 1999)

  60. J.R. Koza, M.A. Keane, M.J. Streeter, W. Mydlowec, J. Yu, G. Lanza, Genetic Programming IV: Routine Human-Competitive Machine Intelligence (Kluwer Academic Publishers, 2003)

  61. O. Landry, Introduction to quantum computing. From Physics Department of McGill University, April 2004 (It appears to be now unavailable on the Internet, but the authors of this paper have a hard copy available upon request)

  62. W.B. Langdon, T. Soule, R. Poli, Foster, J.A. The evolution of size and shape, in Advances in Genetic Programming: Volume 3, ed. by L. Spector, W.B. Langdon, U.-M. O-Reilly, P.J. Angeline (The MIT Press, 1999), pp. 163–190

  63. D. Leibfried, E. Knill, S. Seidelin, J. Britton, R.B. Blakestad, J. Chiaverini1, D.B. Hume, W.M. Itano, J.D. Jost, C. Langer, R. Ozeri, R. Reichle, D.J. Wineland, Creation of a six-atom ’Schrödinger cat’ state. Nature 438(1), 639–642 (December 2005)

    Google Scholar 

  64. A. Leier, Evolution of Quantum Algorithms Using Genetic Programming, PhD thesis, University of Dortmund, Department of Computer Science, 2004

  65. A. Leier, W. Banzhaf, Evolving Hogg’s quantum algorithm using linear-tree GP, in Proceedings of the Genetic and Evolutionary Computation Conference (GECCO–03) (Chicago, 12–16 July 2003), ed. by E. Cantú-Paz, J.A. Foster, K. Deb, D. Davis, R. Roy, U.-M. O’Reilly, H.-G. Beyer, R. Standish, G. Kendall, S. Wilson, M. Harman, J. Wegener, D. Dasgupta, M.A. Potter, A.C. Schultz, K. Dowsland, N. Jonoska, J. Miller. LNCS, vol. 2723 (Springer-Verlag), pp. 390–400

  66. A. Leier, W. Banzhaf, Exploring the search space of quantum programs, in Proceedings of the 2003 Congress on Evolutionary Computation (CEC2003), ed. by R. Sarker, R. Reynolds, H. Abbass, K.C. Tan, B. McKay, D. Essam, T. Gedeon (IEEE Press, December 2003), pp. 170–177

  67. A. Leier, W. Banzhaf, Comparison of selection strategies for evolutionary quantum circuit design, in Proceedings of the Genetic and Evolutionary Computation Conference (GECCO–04). Lecture Notes in Computer Science (Springer, 2004), pp. 557–568

  68. A.K. Lenstra, H.W. Lenstra Jr., The Development of the Number Field Sieve. Lecture Notes in Mathematics, vol. 1554 (Springer-Verlag, 1993)

  69. D. Loss, D. DiVincenzo, Quantum computation with quantum dots. Phys. Rev. A 57, 120–126 (1998)

    Article  Google Scholar 

  70. C. Lu, X. Zhou, O. Gühne, W. Gao, J. Zhang, Z. Yuan, A. Goebel, T. Yang, J. Pan, Experimental entanglement of six photons in graph states. Nature Phys. 3, 91–95 (2007)

    Article  Google Scholar 

  71. M. Lukac, M. Perkowski, Evolving quantum circuits using genetic algorithms, in Proceedings of the 2002 NASA/DOD Conference on Evolvable Hardware (2002), pp. 177–185

  72. M. Lukac, M. Perkowski, Evolutionary approach to quantum symbolic logic synthesis, in Proceedings of the 2008 IEEE Congress on Evolutionary Computation (CEC2008) (June 2008), pp. 3374–3380

  73. M. Lukac, M. Perkowski, P. Kerntopf, M. Pivtoraiko, M. Folgheraiter, D. Lee, H. Kim, W. Hwuangbo, J. wook Kim, Y.W. Choi, A hierarchical approach to computer aided design of quantum circuits, in Proceedings of the 6th International Symposium on Representations and Methodology of Future Computing Technology (2003), pp. 201–209

  74. M. Lukac, M.A. Perkowski, H. Goi, M. Pivtoraiko, C. Hyo Yu, K. Chung, H. Jeech, B. Kim, Y. Kim, Evolutionary approach to quantum and reversible circuits synthesis. Artif. Intell. Rev. 20(3–4), 361–417 (2003)

    Article  MATH  Google Scholar 

  75. S. Luke, L. Spector, A revised comparison of crossover and mutation in genetic programming, in Proceedings of the Third Annual Genetic Programming Conference (Morgan Kaufmann, San Fransisco, 1998), pp. 208–213

  76. P. Massey, J.A. Clark, S. Stepney, Evolving quantum circuits and programs through genetic programming, in Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2004). Lecture Notes in Computer Science, vol. 3103 (Springer, 2004), pp. 569–580. Winner of the “Best in GP Stream” award, GECCO 2004

  77. P. Massey, J.A. Clark, S. Stepney, Evolution of a human-competitive Quantum Fourier Transform algorithm using genetic programming, in Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2005) (ACM Press, 2005), pp. 1657–1664. (Ref. 78 is a revised and extended version of this paper)

  78. P. Massey, J.A. Clark, S. Stepney Human-competitive evolution of quantum computing artefacts by genetic programming. Evol. Comput. J. 14(1), 21–40 (2006) (this is a revised and extended version of Ref. 77)

  79. P.S. Massey, Searching for Quantum Software. PhD thesis, University of York, Department of Computer Science, 2006. Available at http://www.cs.york.ac.uk/ftpdir/reports/YCST-2007-11.pdf

  80. Z. Michalewicz, Genetic algorithms + Data structures = Evolution Programs, 3rd edn. (Springer-Verlag, New York, 1996)

    Google Scholar 

  81. M. Mitchell, An Introduction to Genetic Algorithms (The MIT Press, 1996)

  82. P. Neumann, N. Mizuochi, F. Rempp, P. Hemmer, H. Watanabe, S. Yamasaki, V. Jacques, T. Gaebel, F. Jelezko, J. Wrachtrup, Multipartite entanglement among single spins in diamond. Science 320(5881), 1326–1329 (6 June 2008)

  83. M.A. Nielsen, I.L. Chuang, Quantum computation and Quantum Information (Cambridge University Press, Cambridge 2000)

    MATH  Google Scholar 

  84. T. Pellizzari, Quantum computers: first steps towards a realization, in Non-Standard Computation, ed. by T. Gramß, S. Bornholdt, M. Groß, M. Mitchell, T. Pellizzari (WILEY-VCH, Weinheim, Germany, 1998), pp. 141–178

  85. A.O. Pittenger, An Introduction to Quantum Computing Algorithms (Birkhäuser, Boston 2000)

    MATH  Google Scholar 

  86. G. Păun, G. Rozenberg, A. Salomaa, DNA Computing: New Computing Paradigms (Springer-Verlag, 1998)

  87. G. Păun, G. Rozenberg, A. Salomaa, in DNA Computing: New Computing Paradigms (Springer-Verlag, 1998), pp. 1–6 Introduction; pp. 43–74 Beginnings of Molecular Computing

  88. T. Reid, On the evolutionary design of quantum circuits. Master’s thesis, Waterloo, Ontario, Canada, 2005

  89. M.J. Rethinam, A.K. Javali, E.C. Behrman, J.E. Steck, S.R. Skinner, A genetic algorithm for finding pulse sequences for NMR quantum computing. ArXiv Quantum Physics e-prints quant-ph/0404170v1 (April 2004)

  90. E. Rieffel, W. Polak, An introduction to quantum computing for non-physicists. ACM Comput. Surv. 32(3), 300–335 (2000)

    Article  Google Scholar 

  91. B.I.P. Rubinstein, Evolving quantum circuits using genetic programming, in Proceedings of the 2001 IEEE Congress on Evolutionary Computation (CEC2001) (IEEE Press, 2001), pp. 114–121

  92. A. Sabry, Modeling quantum computing in Haskell, in Haskell ’03: Proceedings of the ACM SIGPLAN workshop on Haskell (ACM Press, New York, NY, USA, 2003), pp. 39–49

  93. Y. Shi, Both Toffoli and Controlled-NOT need little help to do universal quantum computation. ArXiv Quantum Physics e-prints quant-ph/0205115v2 (May 2002)

  94. P.W. Shor, Algorithms for quantum computation: discrete log and factoring, in Proceedings of the 35th Annual Symposium on Foundations of Computer Science (Institute of Electrical and Electronic Engineers Computer Society Press, November 1994), pp. 124–134

  95. P.W. Shor, Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A 52, 2493–2496 (1995)

    Article  Google Scholar 

  96. P.W. Shor, Introduction to quantum algorithms. Notes for talk given for the short course at the January 2000 American Math Society meeting, 2000

  97. D. Simon, On the power of quantum computation, in Proceedings of the 35th annual IEEE symposium on the foundations of computer science (FOCS) (IEEE, Computer Society Press, Santa Fee, USA, November 1994), pp. 116–123

  98. S.F. Smith, A Learning System Based on Genetic Adaptive Algorithms. PhD thesis, University of Pittsburgh, 1980

  99. D.A. Sofge, Prospective algorithms for quantum evolutionary computation, in Proceedings of the Second Quantum Interaction Symposium (QI-2008)(2008). Available from ArXiv e-prints at http://arxiv.org/abs/0804.1133v1

  100. L. Spector, The evolution of arbitrary computational processes. IEEE Intell. Syst. 15(3), 80–83 (May/June 2000)

  101. L. Spector, Automatic Quantum Computer Programming: A Genetic Programming Approach. Genetic Programming Series. (Kluwer Academic Publishers, 2004)

  102. L. Spector, H. Barnum, H.J. Bernstein, Genetic programming for quantum computers, in Genetic Programming 1998: Proceedings of the Thurd Annual Conference, ed. by J.R. Koza, W. Banzhaf, K. Chellapilla, K. Deb, M. Dorigo, D.B. Fogel, M.H. Garzon, D.E. Goldberg, H. Iba, R.L. Riolo (1998), pp. 365–374

  103. L. Spector, H. Barnum, H.J. Bernstein, N. Swamy, Finding a better-than-classical quantum AND/OR algorithm using genetic programming, in Proceedings of the 1999 Congress on Evolutionary Computation (IEEE, 1999), pp. 2239–246

  104. L. Spector, H. Barnum, H.J. Bernstein, N. Swamy, Quantum computing applications of genetic programming, in Advances in Genetic Programming: Volume 3, ed. by L. Spector, W.B. Langdon, U.-M. O-Reilly, P.J. Angeline (The MIT Press, 1999), pp. 135–160

  105. L. Spector, H.J. Bernstein, Communication capacities of some quantum gates, discovered in part through genetic programming, in Proceedings of the 6th International Conference on Quantum Communication, Measurement, and Computing (QCMC), ed. by J.H. Shapiro, O. Hirota (Rinton Press, 2003), pp. 500–503. Preprint available at http://hampshire.edu/lspector/pubs/spector-QCMC-prepress.pdf

  106. L. Spector, W.B. Langdon, U.-M. O-Reilly, P.J. Angeline (eds.), Advances in Genetic Programming: Volume 3 (The MIT Press, 1999)

  107. R. Stadelhofer, Evolving Blackbox Quantum Algorithms using Genetic Programming. PhD thesis, University of Dortmund, Department of Physics (2006)

  108. R. Stadelhofer, W. Banzhaf, D. Suter, Quantum and classical parallelism in parity algorithms for ensemble quantum computers. Phys. Rev. A 71, 032345-1–032345-6 (2005)

  109. R. Stadelhofer, W. Banzhaf, D. Suter, Evolving blackbox quantum algorithms using genetic programming. Artif. Intell. Eng. Des. Anal. Manuf. 22, 285–297 (2008)

    Google Scholar 

  110. S. Stepney, J.A. Clark, Evolving quantum programs and protocols, in Handbook of Theoretical and Computational Nanotechnology, volume 3, Quantum and Molecular Computing, Quantum Simulations, ch. 3, ed. by M. Rieth, W. Schommers (American Scientific Publishers, 2006), pp. 113–160

  111. S. Stepney, J.A. Clark, Searching for quantum programs and quantum protocols: a review. J. Comput. Theor. Nanosci. 5, 942–969 (May, 2008)

    Google Scholar 

  112. P. Surry, N. Radcliffe, A formalism for real-parameter evolutionary algorithms and directed recombination. in Foundations of Genetic Algorithms, vol. 4, ed. by R.K. Belew, M.D. Vose (Morgan Kaufmann, 1997)

  113. A. Teller, M. Veloso, PADO: a new learning architecture for object recognition, in Symbolic Visual Learning, ed. by K. Ikeuchi M. Veloso (Oxford University Press, 1996), pp. 81–116

  114. M. Udrescu, L. Prodan, M. Vlăduţiu, Implementing quantum genetic algorithms: a solution based on grover’s algorithm, in Proceedings of the Third Conference on Computing frontiers (ACM Press, New York, 2006), pp. 71–82

  115. W. van Dam, A universal quantum cellular automaton, in Proceedings of PhysComp96 (New England Complex Systems Institute, 1996), pp. 323–331

  116. R. Van Meter, K. Binkley, Compiling quantum programs using genetic algorithms, in The Wild and Crazy Idea Session IV, abstracts, part of 11th International Conference on Architectural Support for Programming Languages and Operating Systems (October 2004)

  117. A. van Tonder, A lambda calculus for quantum computation. SIAM J. Comput. 33(5), 1109–1135 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  118. L.M.K. Vandersypen, M. Steffen, G. Breyta, C.S. Yannoni, M.H. Sherwood, I.L. Chuang, Experimental realization of Shor’s quantum factoring algorithm using nuclear magnetic resonance. Nature 414, 883–887 (December 2001)

  119. M.D. Vose, The Simple Genetic Algorithm: Foundations and Theory (The MIT Press, 1999)

  120. Watrous, J. Relationships between quantum and classical space-bounded complexity classes, in 13th Annual IEEE Conference on Computational Complexity (June 1998), pp. 210–227

  121. C.P. Williams, S.H. Clearwater, Explorations in Quantum Computing (Springer-Verlag, 1998)

  122. C.P. Williams, A.G. Gray, Automated design of quantum circuits, in QCQC ’98: Selected papers from the First NASA International Conference on Quantum Computing and Quantum Communications, ed. by C.P. Williams. Lecture Notes in Computer Science, vol. 1509 (Springer-Verlag, 1998), pp. 113–125

  123. W.K. Wooters, W.H. Zurek, A single quantum cannot be cloned. Nature 299, 802–803 (1982)

    Google Scholar 

  124. T. Yabuki, H. Iba, Genetic algorithms for quantum circuit design, evolving a simpler teleportation circuit, in Proceedings of the Genetic and Evolutionary Computation Conference (GECCO–00) (Morgan Kauffman Publishers, San Francisco, July 2000), pp. 421–425

  125. Q. Yang, The research of a hybrid quantum-inspired evolutionary algorithm. Master’s thesis, Wuhan University, China, 2006

  126. A. Yao, Quantum circuit complexity, in Proceedings of the 34th Annual Symposium on the Foundations of Computer Science (IEEE Computer Society Press, Los Alamitos, USA, 1993), pp. 352–361

  127. W.H. Zurek, Decoherence and the transition from quantum to classical. Physics Today 44, 36–44 (1991)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Information Technology, Bond University, Robina, QLD, Australia

    Adrian Gepp & Phil Stocks

Authors
  1. Adrian Gepp
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Phil Stocks
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Adrian Gepp.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gepp, A., Stocks, P. A review of procedures to evolve quantum algorithms. Genet Program Evolvable Mach 10, 181–228 (2009). https://doi.org/10.1007/s10710-009-9080-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10710-009-9080-7

Keywords

Search

Navigation