Advertisement

Quantum Information Processing

, Volume 14, Issue 8, pp 2729–2748 | Cite as

Quantum digital-to-analog conversion algorithm using decoherence

  • Akira SaiTohEmail author
Article
First Online:
  • 119 Downloads

Abstract

We consider the problem of mapping digital data encoded on a quantum register to analog amplitudes in parallel. It is shown to be unlikely that a fully unitary polynomial-time quantum algorithm exists for this problem; NP becomes a subset of BQP if it exists. In the practical point of view, we propose a nonunitary linear-time algorithm using quantum decoherence. It tacitly uses an exponentially large physical resource, which is typically a huge number of identical molecules. Quantumness of correlation appearing in the process of the algorithm is also discussed.

Keywords

Digital-to-analog conversion Quantum algorithm  Bulk-ensemble computation 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgments

This work was supported by the Grant-in-Aid for Scientific Research from JSPS (Grant No. 25871052).

Supplementary material

11128_2015_1033_MOESM1_ESM.pdf (37 kb)
Supplementary material 1 (pdf 36 KB)

References

  1. 1.
    Aiello, A., Puentes, G., Woerdman, J.P.: Linear optics and quantum maps. Phys. Rev. A 76, 032323-1-12 (2007)MathSciNetADSCrossRefGoogle Scholar
  2. 2.
    Ali, M., Rau, A.R.P., Alber, G.: Quantum discord for two-qubit x states. Phys. Rev. A 81, 042105-1-7 (2010)ADSGoogle Scholar
  3. 3.
    Almeida, M.P., de Melo, F., Hor-Meyll, M., Salles, A., Walborn, S.P., Souto Ribeiro, P.H., Davidovich, L.: Environment-induced sudden death of entanglement. Science 316, 579–582 (2007)ADSCrossRefGoogle Scholar
  4. 4.
    Arora, S., Barak, B.: Computational Complexity: A Modern Approach. Cambridge University Press, Cambridge (2009)CrossRefGoogle Scholar
  5. 5.
    Bardhan, B.R., Brown, K.L., Dowling, J.P.: Dynamical decoupling with tailored wave plates for long-distance communication using polarization qubits. Phys. Rev. A 88, 052311-1-7 (2013)ADSzbMATHGoogle Scholar
  6. 6.
    Becker, E.D.: High Resolution NMR: Theory and Chemical Applications, 2nd edn. Academic Press, New York (1980)Google Scholar
  7. 7.
    Bengtsson, I., Życzkowski, K.: Geometry of Quantum States: An Introduction to Quantum Entanglement. Cambridge University Press, Cambridge (2006)CrossRefGoogle Scholar
  8. 8.
    Berglund, A.J.: Quantum coherence and control in one- and two-photon optical systems (2000). arXiv:quant-ph/0010001
  9. 9.
    Boykin, P.O., Mor, T., Roychowdhury, V., Vatan, F., Vrijen, R.: Algorithmic cooling and scalable NMR quantum computers. Proc. Natl. Acad. Sci. USA 99(6), 3388–3393 (2002)ADSCrossRefzbMATHGoogle Scholar
  10. 10.
    Brodutch, A., Gilchrist, A., Terno, D.R., Wood, C.J.: Quantum discord in quantum computation. J. Phys.: Conf. Ser. 306, 012030-1-11 (2011)ADSGoogle Scholar
  11. 11.
    Brüschweiler, R.: Novel strategy for database searching in spin liouville space by NMR ensemble computing. Phys. Rev. Lett. 85, 4815–4818 (2000)ADSCrossRefGoogle Scholar
  12. 12.
    Foroozandeh, M., Adams, R.W., Meharry, N.J., Jeannerat, D., Nilsson, M., Morris, G.A.: Ultrahigh-resolution NMR spectroscopy. Angew. Chem. Int. Ed. 53, 6990–6992 (2014)CrossRefGoogle Scholar
  13. 13.
    Freeman, R.: Selective excitation in high-resolution NMR. Chem. Rev. 91, 1397–1412 (1991)CrossRefGoogle Scholar
  14. 14.
    Gruska, J.: Quantum Computing. McGraw-Hill, London (1999)Google Scholar
  15. 15.
    Hamieh, S., Kobes, R., Zaraket, H.: Positive-operator-valued measure optimization of classical correlations. Phys. Rev. A 70, 052325-1-6 (2004)ADSCrossRefGoogle Scholar
  16. 16.
    Henderson, L., Vedral, V.: Classical, quantum and total correlations. J. Phys. A. 34, 6899–6905 (2001)Google Scholar
  17. 17.
    Holevo, A., Giovannetti, V.: Quantum channels and their entropic characteristics. Rep. Prog. Phys. 75, 046001-1-30 (2012)MathSciNetADSCrossRefzbMATHGoogle Scholar
  18. 18.
    Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 81, 865–942 (2009)MathSciNetADSCrossRefzbMATHGoogle Scholar
  19. 19.
    Jeong, Y.C., Lee, J.C., Kim, Y.H.: Experimental implementation of a fully controllable depolarizing quantum operation. Phys. Rev. A 87, 014301-1-4 (2013)ADSCrossRefGoogle Scholar
  20. 20.
    Jones, J.A.: Quantum computing with NMR. Prog. NMR Spectr. 59, 91–120 (2011)CrossRefzbMATHGoogle Scholar
  21. 21.
    Khitrin, A.K., Michalski, M., Lee, J.S.: Reversible projective measurement in quantum ensembles. Quantum Inf. Process. 10, 557–566 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kitagawa, M., Kataoka, A., Nishimura, T.: Initialization and scalability of NMR quantum computers. In: Shapiro, J.H., Hirota, O. (eds.) Proceedings of the 6th International Conference on Quantum Communication, Measurement and Computing (QCMC2002), pp. 275–280. Rinton Press, Princeton (2003)Google Scholar
  23. 23.
    Klenke, A.: Probability Theory: A Comprehensive Course. Springer, London (2008)CrossRefGoogle Scholar
  24. 24.
    Kondo, Y., Nakahara, M., Tanimura, S., Kitajima, S., Uchiyama, C., Shibata, F.: Generation and suppression of decoherence in artificial environment for qubit system. J. Phys. Soc. Jpn. 76, 074002-1-11 (2007)ADSGoogle Scholar
  25. 25.
    Lanyon, B.P., Barbieri, M., Almeida, M.P., White, A.G.: Experimental quantum computing without entanglement. Phys. Rev. Lett. 101, 200501-1-4 (2008)ADSCrossRefGoogle Scholar
  26. 26.
    Li, C.K., Roberts, R., Yin, X.: Decomposition of unitary matrices and quantum gates. Int. J. Quantum Inf. 11, 130015-1-10 (2013)MathSciNetGoogle Scholar
  27. 27.
    Long, G.L., Xiao, L.: Experimental realization of a fetching algorithm in a 7-qubit NMR spin Liouville space computer. J. Chem. Phys. 119, 8473–8481 (2003)ADSCrossRefzbMATHGoogle Scholar
  28. 28.
    Luo, S.: Quantum discord for two-qubit systems. Phys. Rev. A 77, 042303-1-6 (2008)ADSGoogle Scholar
  29. 29.
    Mehring, M., Weberruß, V.A.: Object-Oriented Magnetic Resonance. Academic Press, San Diego (2001)Google Scholar
  30. 30.
    Microchip Technology Inc.: MCP4725 Data Sheet (2007); ibid. MCP4801/4811/4821 Data Sheet (2010)Google Scholar
  31. 31.
    Milburn, G.J., White, A.G.: Quantum computing using optics. In: Meyers, R.A. (ed.) Computational Complexity, pp. 2437–2452. Springer, New York (2012)CrossRefGoogle Scholar
  32. 32.
    Morris, G.A., Freeman, R.: Selective excitation in Fourier transform nuclear magnetic resonance. J. Magn. Res. 29, 433–462 (1978)ADSGoogle Scholar
  33. 33.
    Nakajima, Y., Kawano, Y., Sekigawa, H.: A new algorithm for producing quantum circuits using KAK decompositions. Quantum Inf. Comput. 6, 067–080 (2006)MathSciNetGoogle Scholar
  34. 34.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  35. 35.
    Ollivier, H., Zurek, W.H.: Quantum discord: a measure of the quantumness of correlations. Phys. Rev. Lett. 88, 017901-1-4 (2001)ADSCrossRefGoogle Scholar
  36. 36.
    Ortigosa-Blanch, A., Capmany, J.: Subcarrier multiplexing optical quantum key distribution. Phys. Rev. A 73, 024305-1-4 (2006)ADSCrossRefGoogle Scholar
  37. 37.
    Peters, N., Altepeter, J., Jeffrey, E., Branning, D., Kwiat, P.: Precise creation, characterization and manipulation of single optical qubits. Quantum Inf. Comput. 3, 503–517 (2003)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Plenio, M.B., Virmani, S.: An introduction to entanglement measures. Quantum Inf. Comput. 7, 1–51 (2007)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Radulov, G., Quinn, P., Hegt, H., van Roermund, A.: Smart and Flexible Digital-to-Analog Converters. Springer, Heidelberg (2011)CrossRefzbMATHGoogle Scholar
  40. 40.
    Ramelow, S., Ratschbacher, L., Fedrizzi, A., Langford, N.K., Zeilinger, A.: Discrete tunable color entanglement. Phys. Rev. Lett. 103, 253601 (2009)ADSCrossRefGoogle Scholar
  41. 41.
    SaiToh, A., Kitagawa, M.: Numerical analysis of boosting scheme for scalable NMR quantum computation. Phys. Rev. A 71, 022303-1-13 (2005)ADSCrossRefGoogle Scholar
  42. 42.
    SaiToh, A., Kitagawa, M.: Matrix-product-state simulation of an extended Brüschweiler bulk-ensemble database search. Phys. Rev. A 73, 062332-1-19 (2006)ADSCrossRefGoogle Scholar
  43. 43.
    SaiToh, A., Rahimi, R., Nakahara, M.: Limitation for linear maps in a class for detection and quantification of bipartite nonclassical correlation. Quantum Inf. Comput. 12, 0944–0952 (2012)MathSciNetGoogle Scholar
  44. 44.
    Schmüser, F., Janzing, D.: Quantum analog-to-digital and digital-to-analog conversion. Phys. Rev. A 72, 042324-1-8 (2005)ADSCrossRefzbMATHGoogle Scholar
  45. 45.
    Schulman, L.J., Vazirani, U.V.: Molecular scale heat engines and scalable quantum computation. In: Proceedings of 31st Annual ACM Symposium on Theory of Computing (STOC99), pp. 322–329. ACM, New York (1999)Google Scholar
  46. 46.
    Shende, V.V., Bullock, S.S., Markov, I.L.: Synthesis of quantum logic circuits. IEEE TCAD 25, 1000–1010 (2006)Google Scholar
  47. 47.
    Shiryaev, A.N.: Probability, 2nd edn. Springer, New York (1996). Translated by R.P. BoasGoogle Scholar
  48. 48.
    Tarasov, V.E.: Quantum computer with mixed states and four-valued logic. J. Phys. A: Math. Gen. 35, 5207–5235 (2002)MathSciNetADSCrossRefGoogle Scholar
  49. 49.
    Tucci, R.R.: A rudimentary quantum compiler (1999). arXiv:quant-ph/9902062
  50. 50.
    Usmani, I., Afzelius, M., de Riedmatten, H., Gisin, N.: Mapping multiple photonic qubits into and out of one solid-state atomic ensemble. Nat. Commun. 1, 12-1-7 (2010)ADSCrossRefGoogle Scholar
  51. 51.
    Vartiainen, J.J., Möttönen, M., Salomaa, M.M.: Efficient decomposition of quantum gates. Phys. Rev. Lett. 92, 177902-1-4 (2004)ADSCrossRefGoogle Scholar
  52. 52.
    Ventura, D., Martinez, T.: Initializing the amplitude distribution of a quantum state. Found. Phys. Lett. 12, 547–559 (1999)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Watrous, J.: Quantum computational complexity. In: Meyers, R.A. (ed.) Encyclopedia of Complexity and Systems Science, pp. 1–40. Springer, New York (2009, 2014). arXiv:0804.3401
  54. 54.
    Xiao, L., Jones, J.A.: NMR analogues of the quantum Zeno effect. Phys. Lett. A 359, 424–427 (2006)ADSCrossRefGoogle Scholar
  55. 55.
    Xiao, L., Long, G.L.: Fetching marked items form an unsorted database in NMR ensemble computing. Phys. Rev. A 66, 052320-1-5 (2002)ADSCrossRefzbMATHGoogle Scholar
  56. 56.
    Zhang, T., Yin, Z.Q., Han, Z.F., Guo, G.C.: A frequency-coded quantum key distribution scheme. Opt. Commun. 281, 4800–4802 (2008)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Computer Science and EngineeringToyohashi University of TechnologyToyohashiJapan

Personalised recommendations

Cite article

Buy options