On the role of dealing with quantum coherence in amplitude amplification

  • Alexey E. RasteginEmail author


Amplitude amplification is one of primary tools in building algorithms for quantum computers. This technique generalizes key ideas of the Grover search algorithm. Potentially useful modifications are connected with changing phases in the rotation operations and replacing the intermediate Hadamard transform with arbitrary unitary one. In addition, arbitrary initial distribution of the amplitudes may be prepared. We examine trade-off relations between measures of quantum coherence and the success probability in amplitude amplification processes. As measures of coherence, the geometric coherence and the relative entropy of coherence are considered. In terms of the relative entropy of coherence, complementarity relations with the success probability seem to be the most expository. The general relations presented are illustrated within several model scenarios of amplitude amplification processes.


Grover’s algorithm Quantum search Relative entropy Geometric coherence 


  1. 1.
    Grover, L.K.: Quantum mechanics helps in searching for a needle in a haystack. Phys. Rev. Lett. 79, 325–328 (1997)ADSCrossRefGoogle Scholar
  2. 2.
    Grover, L.K.: Quantum computers can search arbitrarily large databases by a single query. Phys. Rev. Lett. 79, 4709–4712 (1997)ADSCrossRefGoogle Scholar
  3. 3.
    Grover, L.K.: Quantum computers can search rapidly by using almost any transformation. Phys. Rev. Lett. 80, 4329–4332 (1998)ADSCrossRefGoogle Scholar
  4. 4.
    Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26, 1484–1509 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Haase, D., Maier, H.: Quantum algorithms for number fields. Fortschr. Phys. 54, 866–881 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hallgren, S.: Polynomial-time quantum algorithms for Pell’s equation and the principal ideal problem. J. ACM 54, 4 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Childs, A.M., van Dam, W.: Quantum algorithms for algebraic problems. Rev. Mod. Phys. 82, 1–52 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Lomonaco, S.J., Kauffman, L.H.: Is Grover’s algorithm a quantum hidden subgroup algorithm? Quantum Inf. Process. 6, 461–476 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bennett, C.H., Bernstein, E., Brassard, G., Vazirani, U.: Strengths and weaknesses of quantum computing. SIAM J. Comput. 26, 1510–1523 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Zalka, C.: Grover’s quantum searching algorithm is optimal. Phys. Rev. A 60, 2746–2751 (1999)ADSCrossRefGoogle Scholar
  11. 11.
    Patel, A.D., Grover, L.K.: Quantum search. In: Kao, M.-Y. (ed.) Encyclopedia of Algorithms, pp. 1707–1716. Springer, New York (2016)CrossRefGoogle Scholar
  12. 12.
    Biham, E., Biham, O., Biron, D., Grassl, M., Lidar, D.A.: Grover’s quantum search algorithm for an arbitrary initial amplitude distribution. Phys. Rev. A 60, 2742–2745 (1999)ADSCrossRefzbMATHGoogle Scholar
  13. 13.
    Biham, E., Biham, O., Biron, D., Grassl, M., Lidar, D.A., Shapira, D.: Analysis of generalized Grover quantum search algorithms using recursion equations. Phys. Rev. A 63, 012310 (2000)ADSCrossRefGoogle Scholar
  14. 14.
    Biham, E., Kenigsberg, D.: Grover’s quantum search algorithm for an arbitrary initial mixed state. Phys. Rev. A 66, 062301 (2002)ADSCrossRefGoogle Scholar
  15. 15.
    Deutsch, D.: Quantum theory, the Church-Turing principle and the universal quantum computer. Proc. R. Soc. Lond. A 400, 97–117 (1985)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Braunstein, S.L., Pati, A.K.: Speed-up and entanglement in quantum searching. Quantum Inf. Comput. 2, 399–409 (2002)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Jozsa, R., Linden, N.: On the role of entanglement in quantum-computational speed-up. Proc. R. Soc. Lond. A 459, 2011–2032 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Baumgratz, T., Cramer, M., Plenio, M.B.: Quantifying coherence. Phys. Rev. Lett. 113, 140401 (2014)ADSCrossRefGoogle Scholar
  19. 19.
    Adesso, G., Bromley, T.R., Cianciaruso, M.: Measures and applications of quantum correlations. J. Phys. A: Math. Theor. 49, 473001 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Streltsov, A., Adesso, G., Plenio, M.B.: Quantum coherence as a resource. Rev. Mod. Phys. 89, 041003 (2017)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Hu, M.-L., Hu, X., Peng, Y., Zhang, Y.-R., Fan, H.: Quantum coherence and quantum correlations. E-print arXiv:1703.01852 [quant-ph] (2017)
  22. 22.
    Hillery, M.: Coherence as a resource in decision problems: the Deutsch-Jozsa algorithm and a variation. Phys. Rev. A 93, 012111 (2016)ADSCrossRefGoogle Scholar
  23. 23.
    Shi, H.-L., Liu, S.-Y., Wang, X.-H., Yang, W.-L., Yang, Z.-Y., Fan, H.: Coherence depletion in the Grover quantum search algorithm. Phys. Rev. A 95, 032307 (2017)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Anand, N., Pati, A.K.: Coherence and entanglement monogamy in the discrete analogue of analog Grover search. E-print arXiv:1611.04542 [quant-ph] (2016)
  25. 25.
    Farhi, E., Gutmann, S.: Analog analogue of a digital quantum computation. Phys. Rev. A 57, 2403–2406 (1998)ADSCrossRefGoogle Scholar
  26. 26.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  27. 27.
    Watrous, J.: The Theory of Quantum Information. Cambridge University Press, Cambridge (2018)CrossRefzbMATHGoogle Scholar
  28. 28.
    Vedral, V.: The role of relative entropy in quantum information theory. Rev. Mod. Phys. 74, 197–234 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Singh, U., Pati, A.K., Bera, M.N.: Uncertainty relations for quantum coherence. Mathematics 4, 47 (2016)CrossRefzbMATHGoogle Scholar
  30. 30.
    Peng, Y., Zhang, Y.-R., Fan, Z.-Y., Liu, S., Fan, H.: Complementary relation of quantum coherence and quantum correlations in multiple measurements. E-print arXiv:1608.07950 [quant-ph] (2016)
  31. 31.
    Rastegin, A.E.: Uncertainty relations for quantum coherence with respect to mutually unbiased bases. Front. Phys. 13, 130304 (2018)CrossRefGoogle Scholar
  32. 32.
    Rastegin, A.E.: Quantum coherence quantifiers based on the Tsallis relative \(\alpha \) entropies. Phys. Rev. A 93, 032136 (2016)ADSCrossRefGoogle Scholar
  33. 33.
    Shao, L.-H., Li, Y., Luo, Y., Xi, Z.: Quantum coherence quantifiers based on the Rényi \(\alpha \)-relative entropy. Commun. Theor. Phys. 67, 631–636 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Streltsov, A., Kampermann, H., Wölk, S., Gessner, M., Bruß, D.: Maximal coherence and the resource theory of purity. New J. Phys. 20, 053058 (2018)ADSCrossRefGoogle Scholar
  35. 35.
    Cheng, S., Hall, M.J.W.: Complementarity relations for quantum coherence. Phys. Rev. A 92, 042101 (2015)ADSCrossRefGoogle Scholar
  36. 36.
    Yuan, X., Bai, G., Peng, T., Ma, X.: Quantum uncertainty relation using coherence. Phys. Rev. A 96, 032313 (2017)ADSMathSciNetCrossRefGoogle Scholar
  37. 37.
    Hu, M.-L., Fan, H.: Evolution equation for quantum coherence. Sci. Rep. 6, 29260 (2016)ADSCrossRefGoogle Scholar
  38. 38.
    Shao, L.-H., Xi, Z., Fan, H., Li, Y.: Fidelity and trace-norm distances for quantifying coherence. Phys. Rev. A 91, 042120 (2015)ADSCrossRefGoogle Scholar
  39. 39.
    Rana, S., Parashar, P., Lewenstein, M.: Trace-distance measure of coherence. Phys. Rev. A 93, 012110 (2016)ADSMathSciNetCrossRefGoogle Scholar
  40. 40.
    Uhlmann, A.: The ’transition probability’ in the state space of a *-algebra. Rep. Math. Phys. 9, 273–279 (1976)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Jozsa, R.: Fidelity for mixed quantum states. J. Mod. Optics 41, 2315–2323 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Zhang, H.-J., Chen, B., Li, M., Fei, S.-M., Long, G.-L.: Estimation on geometric measure of quantum coherence. Commun. Theor. Phys. 67, 166–170 (2017)ADSCrossRefzbMATHGoogle Scholar
  43. 43.
    Bera, M.N., Qureshi, T., Siddiqui, M.A., Pati, A.K.: Duality of quantum coherence and path distinguishability. Phys. Rev. A 92, 012118 (2015)ADSCrossRefGoogle Scholar
  44. 44.
    Bagan, E., Bergou, J.A., Cottrell, S.S., Hillery, M.: Relations between coherence and path information. Phys. Rev. Lett. 116, 160406 (2016)ADSCrossRefGoogle Scholar
  45. 45.
    Qureshi, T., Siddiqui, M.A.: Wave-particle duality in \(N\)-path interference. Ann. Phys. 385, 598–604 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Hu, M.-L., Fan, H.: Relative quantum coherence, incompatibility and quantum correlations of states. Phys. Rev. A 95, 052106 (2017)ADSCrossRefGoogle Scholar
  47. 47.
    Ambainis, A., Schulman, L.J., Vazirani, U.: Computing with highly mixed states. J. ACM 53, 507–531 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Popescu, P., Sluşanschi, E.-I., Iancu, V., Pop, F.: A new upper bound for Shannon entropy. A novel approach in modeling of Big Data applications. Concurrency Computat. 28, 351–359 (2016)CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Theoretical PhysicsIrkutsk State UniversityIrkutskRussia

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