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Quantum Gates

  • Chapter
Explorations in Quantum Computing

Part of the book series: Texts in Computer Science ((TCS))

Abstract

In this chapter we introduced the idea of a quantum gate, and contrasted it with logically irreversible and logically reversible classical gates. Quantum gates are, like classical reversible gates, logically reversible, but they differ markedly on their universality properties. Whereas the smallest universal classical reversible gates have to use three bits, the smallest universal quantum gates need only use two bits. As the classical reversible gates are also unitary, it is conceivable that one of the first practical applications of quantum gates is in non-standard (e.g., “spintronic”) implementations of classical reversible computers.

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Notes

  1. 1.

    Source: Opening words of the “Atoms in a SmallWorld” section of Richard Feynman’s classic talk “There’s Plenty of Room at the Bottom,” given on 29th December 1959 at the annual meeting of the American Physical Society at the California Institute of Technology. The full transcript of the talk is available at http://www.zyvex.com/nanotech/feynman.html.

  2. 2.

    What we call an “ancilla bit” is also referred to as a “storage bit” or a “garbage bit” in the literature.

  3. 3.

    A balanced function on {0,1}n returns a value “1” for 2n−1 of its inputs and a value “0” for the other 2n−1 inputs.

  4. 4.

    If A and B are two matrices B is the inverse of A when A.B = 1 where 1 is the identity matrix, i.e., a matrix having only ones down the main diagonal.

  5. 5.

    N.B. the leading “1” in the series expansion of the exponential function is replaced with the identity matrix, 1.

References

  1. A. Barenco, “A Universal Two-Bit Gate for Quantum Computation,” Proc. R. Soc. Lond. A, Volume 449, Issue 1937 (1995) pp. 679–683.

    Article  MathSciNet  MATH  Google Scholar 

  2. A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter, “Elementary Gates for Quantum Computation,” Phys. Rev. A, Volume 52 (1995) pp. 3457–3467.

    Article  Google Scholar 

  3. C. Bennett, “Logical Reversibility of Computation,” IBM Journal of Research and Development, Volume 17 (1973) pp. 525–532.

    Article  MATH  Google Scholar 

  4. C. Bennett, “Time-Space Tradeoff for Reversible Computation,” SIAM Journal on Computing, Volume 18 (1989) pp. 766–776.

    Article  MathSciNet  MATH  Google Scholar 

  5. C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, “Mixed-state Entanglement and Quantum Error Correction,” Phys. Rev. A, Volume 54, Issue 5 (1996) pp. 3824–3851.

    Article  MathSciNet  Google Scholar 

  6. G. Brassard, S. L. Braunstein, and R. Cleve, “Teleportation as a Quantum Computation,” Physica D, Volume 120, Issues 1–2 (1998) pp. 43–47.

    MathSciNet  MATH  Google Scholar 

  7. M. J. Bremner, C. M. Dawson, J. L. Dodd, A. Gilchrist, A. W. Harrow, D. Mortimer, M. A. Nielsen, and T. J. Osborne, “Practical Scheme for Quantum Computation with Any Two-Qubit Entangling Gate,” Phys. Rev. Lett., Volume 89 (2002) 247902.

    Article  Google Scholar 

  8. H. Buhrman, J. Tromp, and P. Vitányi, “Time and Space Bounds for Reversible Simulation,” Journal of Physics A: Mathematical & General, Volume 34, Issue 35 (2001) pp. 6821–6830.

    Article  MATH  Google Scholar 

  9. S. Bullock and I. Markov, “Arbitrary Two-qubit Computation in 23 Elementary Gates,” Phys. Rev. A, Volume 68 (2003) 012318.

    Article  Google Scholar 

  10. R. Cleve, “Methodologies for Designing Block Ciphers and Cryptographic Protocols,” Ph. D. Thesis, University of Toronto (1990).

    Google Scholar 

  11. R. Cleve, “Complexity Theoretic Issues Concerning Block Ciphers Related to DES,” in Proceedings of Advances in Cryptology (CRYPTO’90), Springer, Berlin (1991) pp. 530–544.

    Google Scholar 

  12. P. Crescenzi and C. H. Papadimitriou, “Reversible Simulation of Space-bounded Computations,” Theoretical Computer Science, Volume 143, Issue 1 (1995) pp. 159–165.

    MathSciNet  MATH  Google Scholar 

  13. D. Deutsch, “Quantum Computational Networks,” Proc. R. Soc. Lond. A, Volume 425, Issue 1868 (1989) pp. 73–90.

    Article  MathSciNet  MATH  Google Scholar 

  14. D. P. DiVincenzo, “Two-Bit Gates are Universal for Quantum Computation,” Phys. Rev. A, Volume 51 (1995) pp. 1015–1022.

    Article  Google Scholar 

  15. D. P. DiVincenzo and J. Smolin, “Results on Two-Bit Gate Design for Quantum Computers,” in Proceedings of the Workshop on Physics and Computation, IEEE Computer Society Press, Dallas (1994) pp. 14–23.

    Chapter  Google Scholar 

  16. E. Fredkin and T. Toffoli, “Conservative Logic,” International Journal of Theoretical Physics, Volume 21 (1982) pp. 219–253.

    Article  MathSciNet  MATH  Google Scholar 

  17. R. B. Griffiths and C. Niu, “Semiclassical Fourier Transform for Quantum Computation,” Phys. Rev. Lett., Volume 76 (1996) pp. 3228–3231.

    Article  Google Scholar 

  18. S. Hill and W. Wootters, “Entanglement of a Pair of Quantum Bits,” Phys. Rev. Lett., Volume 78, Issue 26 (1997) pp. 5022–5025.

    Article  Google Scholar 

  19. N. Khaneja, R. Brockett, and S. J. Glaser, “Time Optimal Control in Spin Systems,” Phys. Rev. A, Volume 63 (2001) 032308.

    Article  Google Scholar 

  20. A. Y. Kitaev, “Quantum Computations: Algorithms and Error Correction,” Russ. Math. Surv., Volume 52, Issue 6 (1997) pp. 1191–1249.

    Article  MathSciNet  MATH  Google Scholar 

  21. B. Kraus and J. I. Cirac, “Optimal Creation of Entanglement Using a Two-Qubit Gate,” Phys. Rev. A, Volume 63 (2001) 062309.

    Article  MathSciNet  Google Scholar 

  22. R. Landauer, “Irreversibility and Heat Generation in the Computing Process,” IBM Journal of Research and Development, Volume 5 (1961) pp. 183–191.

    Article  MathSciNet  MATH  Google Scholar 

  23. K. J. Lang, P. McKenzie, and A. Tapp, “Reversible Space Equals Deterministic Space,” Journal of Computers and System Science, Volume 60 (2000) pp. 354–367.

    Article  Google Scholar 

  24. R. Y. Levine and A. T. Sherman, “A Note on Bennett’s Time-space Tradeoff for Reversible Computation,” SIAM Journal on Computing, Volume 19 (1990) pp. 673–677.

    Article  MathSciNet  MATH  Google Scholar 

  25. M. Li, J. Tromp, and P. Vitányi, “Reversible Simulation of Irreversible Computation,” Physica D, Volume 120 (1998) pp. 168–176.

    Article  Google Scholar 

  26. D. Maslov and G. W. Dueck, “Reversible Cascades with Minimal Garbage,” IEEE Transactions on Computer-Aided Design of Integrated Circuits & Systems, Volume 23, Issue 11 (2004) pp. 1497–1509.

    Article  Google Scholar 

  27. V. V. Shende, A. K. Prasad, I. L. Markov, and J. P. Hayes, “Synthesis of Reversible Logic Circuits,” IEEE Transactions on Computer-Aided Design of Integrated Circuits & Systems, Volume 22, Issue 6 (2003) pp. 710–722.

    Article  Google Scholar 

  28. V. V. Shende, I. L. Markov, and S. S. Bullock, “Minimal Universal Two-qubit Controlled-NOT-based Circuits,” Phys. Rev. A, Volume 69 (2004) 062321.

    Article  Google Scholar 

  29. T. Toffoli, “Reversible Computing,” in Proceedings of Automata, Languages and Programming, Seventh Colloquium, ed. de Bakker, Springer, Berlin (1980) pp. 632–644.

    Chapter  Google Scholar 

  30. F. Vatan and C. P. Williams, “Optimal Quantum Circuits for General Two-qubit Gates,” Phys. Rev. A, Volume 69 (2004) 032315.

    Article  Google Scholar 

  31. G. Vidal and C. M. Dawson, “Universal Quantum Circuit for Two-qubit Transformations with Three Controlled-NOT Gates,” Phys. Rev. A, Volume 69 (2004) 010301(R).

    Google Scholar 

  32. P. Zanardi, C. Zalka, and L. Faoro, “Entangling Power of Quantum Evolutions,” Phys. Rev. A, Volume 62 (2000) 30301(R).

    Article  MathSciNet  Google Scholar 

  33. J. Zhang, J. Vala, S. Sastry, and K. B. Whaley, “Geometric Theory of Nonlocal Two-qubit Operations,” Phys. Rev. A, Volume 67 (2003) 042313.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. California Institute of Technology, NASA Jet Propulsion Laboratory, Oak Grove Drive 4800, Pasadena, CA, 91109-8099, USA

    Dr. Colin P. Williams

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  1. Dr. Colin P. Williams
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Correspondence to Colin P. Williams .

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Williams, C.P. (2011). Quantum Gates. In: Explorations in Quantum Computing. Texts in Computer Science. Springer, London. https://doi.org/10.1007/978-1-84628-887-6_2

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  • DOI: https://doi.org/10.1007/978-1-84628-887-6_2

  • Publisher Name: Springer, London

  • Print ISBN: 978-1-84628-886-9

  • Online ISBN: 978-1-84628-887-6

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