Skip to main content

Advertisement

Springer Nature Link
Log in

Decoherence in Grover search algorithm

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

Grover search algorithm provides a quadratic speedup over all classical algorithms for unstructured data search. The origin of this quantum advantage has been explored and studied from various angles. In this work, we investigate this issue from the perspective of decoherence induced by the quantum channel associated to Grover search algorithm. We establish a complementary relation between coherence and success probability for the generalized Grover search algorithm with the register initialized in arbitrary pure state or pseudo-pure state. We provide an operational illustration of decoherence as success probability for the Grover search algorithm with unital noise and the register initialized in the maximal superposition state, which includes standard Grover search algorithm as a special case. To understand the dynamic of decoherence and its relation with success probability intuitively, we evaluate it for the Grover search algorithm with bit flip noise, phase flip noise, amplitude damping noise, and phase damping noise, and observe that both decoherence and success probability exhibit similar behaviors with the time of iterations.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

Data availability

No datasets were generated or analysed during the current study.

References

  1. Baumgratz, T., Cramer, M., Plenio, M.B.: Quantifying coherence. Phys. Rev. Lett. 113, 140401 (2014)

    ADS  Google Scholar 

  2. Yu, X.D., Zhang, D.J., Xu, G.F., Tong, D.M.: Alternative framework for quantifying coherence. Phys. Rev. A 94, 060302(R) (2016)

    ADS  Google Scholar 

  3. Streltsov, A., Adesso, G., Plenio, M.B.: Quantum coherence as a resource. Rev. Mod. Phys. 89, 041003 (2017)

    ADS  MathSciNet  Google Scholar 

  4. Hu, M.-L., Hu, X., Wang, J., Peng, Y., Zhang, Y.-R., Fan, H.: Quantum coherence and geometric quantum discord. Phys. Rep. 762–764, 1 (2018)

    ADS  MathSciNet  Google Scholar 

  5. Luo, S., Sun, Y.: Coherence and complementarity in state-channel interaction. Phys. Rev. A 98, 012113 (2018)

    ADS  Google Scholar 

  6. Bischof, F., Kampermann, H., Bruß, D.: Resource theory of coherence based on positive-operator-valued measures. Phys. Rev. Lett. 123, 110402 (2019)

    ADS  MathSciNet  Google Scholar 

  7. Xu, J., Shao, L.H., Fei, S.M.: Coherence measures with respect to general quantum measurements. Phys. Rev. A 102, 012411 (2020)

    ADS  MathSciNet  Google Scholar 

  8. Bischof, F., Kampermann, H., Bruß, D.: Quantifying coherence with respect to general quantum measurements. Phys. Rev. A 103, 032429 (2021)

    ADS  MathSciNet  Google Scholar 

  9. Theurer, T., Killoran, N., Egloff, D., Plenio, M.B.: Resource theory of superposition. Phys. Rev. Lett. 119, 230401 (2017)

    ADS  Google Scholar 

  10. Das, S., Mukhopadhyay, C., Roy, S.S., Bhattacharya, S., Sen(De), A., Sen, U.: Wave-particle duality employing quantum coherence in superposition with non-orthogonal pointers. J. Phys. A Math. Theor. 53, 115301 (2020)

  11. Torun, G., Şenyaşa, H.T., Yildiz, A.: Resource theory of superposition: state transformations. Phys. Rev. A 103, 032416 (2021)

    ADS  MathSciNet  Google Scholar 

  12. Ringbauer, M., Bromley, T.R., Cianciaruso, M., Lami, L., Lau, W.Y.S., Adesso, G., White, A.G., Fedrizzi, A., Piani, M.: Certification and quantification of multilevel quantum coherence. Phys. Rev. X 8, 041007 (2018)

    Google Scholar 

  13. Johnston, N., Li, C.-K., Plosker, S., Poon, Y.-T., Regula, B.: Evaluating the robustness of \(k\)-coherence and \(k\)-entanglement. Phys. Rev. A 98, 022328 (2018)

    ADS  Google Scholar 

  14. Regula, B., Piani, M., Cianciaruso, M., Bromley, T.R., Streltsov, A., Adesso, G.: Converting multilevel nonclassicality into genuine multipartite entanglement. New J. Phys. 20, 033012 (2018)

    ADS  Google Scholar 

  15. Johnston, N., Moein, S., Pereira, R., Plosker, S.: Absolutely \(k\)-incoherent quantum states and spectral inequalities for the factor width of a matrix. Phys. Rev. A 106, 052417 (2022)

    ADS  MathSciNet  Google Scholar 

  16. Designolle, S., Uola, R., Luoma, K., Brunner, N.: Set coherence: basis-independent quantification of quantum coherence. Phys. Rev. Lett. 126, 220404 (2021)

    ADS  MathSciNet  Google Scholar 

  17. Ahnefeld, F., Theurer, T., Egloff, D., Matera, J.M., Plenio, M.B.: Coherence as a Resource for Shor’s Algorithm. Phys. Rev. Lett. 129, 120501 (2022)

    ADS  MathSciNet  Google Scholar 

  18. Anand, N., Pati, A.K.: Coherence and entanglement monogamy in the discrete analogue of analog Grover search. arXiv:1611.04542 (2016)

  19. 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)

    ADS  MathSciNet  Google Scholar 

  20. Chin, S.: Coherence number as a discrete quantum resource. Phys. Rev. A 96, 042336 (2017)

    ADS  Google Scholar 

  21. Rastegin, A.E.: Degradation of Grover’s search under collective phase flips in queries to the oracle. Front. Phys. 13, 130318 (2018)

    Google Scholar 

  22. Rastegin, A.E.: On the role of dealing with quantum coherence in amplitude amplification. Quant. Inf. Process. 17, 179 (2018)

    ADS  MathSciNet  Google Scholar 

  23. Pan, M., Qiu, D.: Operator coherence dynamics in Grover’s quantum search algorithm. Phys. Rev. A 100, 012349 (2019)

    ADS  MathSciNet  Google Scholar 

  24. Liu, Y.C., Shang, J., Zhang, X.: Coherence depletion in quantum algorithms. Entropy 21, 260 (2019)

    ADS  MathSciNet  Google Scholar 

  25. Pan, M., Situ, H., Zheng, S.: Complementarity between success probability and coherence in Grover search algorithm. Europhys. Lett. 138, 48002 (2022)

    ADS  Google Scholar 

  26. Ye, L., Wu, Z., Fei, S.M.: Tsallis relative \(\alpha \) entropy of coherence dynamics in Grover’s search algorithm. Commun. Theor. Phys. 75, 085101 (2023)

    ADS  MathSciNet  Google Scholar 

  27. Rastegin, A.E., Anzhelika, M.S.: Degeneration of the Grover search algorithm with depolarization in the oracle-box wires. Mod. Phys. Lett. A 38, 2350030 (2023)

    ADS  MathSciNet  Google Scholar 

  28. Hillery, M.: Coherence as a resource in decision problems: The Deutsch-Jozsa algorithm and a variation. Phys. Rev. A 93, 012111 (2016)

    ADS  Google Scholar 

  29. Naseri, M., Kondra, T.V., Goswami, S., Fellous-Asiani, M., Streltsov, A.: Entanglement and coherence in the Bernstein-Vazirani algorithm. Phys. Rev. A 106, 062429 (2022)

    ADS  MathSciNet  Google Scholar 

  30. Feng, C., Chen, L., Zhao, L.J.: Coherence and entanglement in Grover and Harrow-Hassidim-Lloyd algorithm. Phys. A 626, 129048 (2023)

    MathSciNet  Google Scholar 

  31. Grover, L.K.: Quantum mechanics helps in searching for a needle in a haystack. Phys. Rev. Lett. 79, 325 (1997)

    ADS  Google Scholar 

  32. Grover, L.K.: Quantum computers can search arbitrarily large databases by a single query. Phys. Rev. Lett. 79, 4709 (1997)

    ADS  Google Scholar 

  33. Grover, L.K.: Quantum computers can search rapidly by using almost any transformation. Phys. Rev. Lett. 80, 4329 (1998)

    ADS  Google Scholar 

  34. Bennett, C.H., Bernstein, E., Brassard, G., Vazirani, U.: Strengths and weaknesses of quantum computing SIAM. J. Comput. 26, 1510 (1997)

    MathSciNet  Google Scholar 

  35. Zalka, C.: Grover’s quantum searching algorithm is optimal. Phys. Rev. A 60, 4 (1999)

    Google Scholar 

  36. Long, G., Li, Y., Zhang, W., Niu, L.: Phase matching in quantum searching. Phys. Lett. A 262, 27 (1999)

    ADS  MathSciNet  Google Scholar 

  37. Long, G.L., Li, Y.S., Xiao, L., et al.: Phase matching in quantum searching and the improved Grover algorithm. Nucl. Phys. Rev. 21, 114 (2004)

    Google Scholar 

  38. 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 (1999)

    ADS  Google Scholar 

  39. 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)

    ADS  Google Scholar 

  40. Biham, E., Kenigsberg, D.: Grover’s quantum search algorithm for an arbitrary initial mixed state. Phys. Rev. A 66, 062301 (2002)

    ADS  Google Scholar 

  41. Biham, O., Shapira, D., Shimoni, Y.: Analysis of Grover’s quantum search algorithm as a dynamical system. Phys. Rev. A 68, 022326 (2003)

    ADS  Google Scholar 

  42. Shapira, D., Shimoni, Y., Biham, O.: Algebraic analysis of quantum search with pure and mixed states. Phys. Rev. A 71, 042320 (2005)

    ADS  MathSciNet  Google Scholar 

  43. Yoder, T., Low, G.H., Chuang, I.: Fixed-point quantum search with an optimal number of queries. Phys. Rev. Lett. 113, 210501 (2014)

    ADS  Google Scholar 

  44. Tulsi, A.: Faster quantum searching with almost arbitrary operators. Phys. Rev. A 91, 052307 (2015)

    ADS  Google Scholar 

  45. Roy, T., Jiang, L., Schuster, D.I.: Deterministic Grover search with a restricted oracle. Phys. Rev. Res. 4, L022013 (2022)

    Google Scholar 

  46. Galindo, A., Martin-Delgado, M.A.: Family of Grover’s quantum searching algorithms. Phys. Rev. A 62, 062303 (2000)

    ADS  Google Scholar 

  47. Shapira, D., Mozes, S., Biham, O.: Effect of unitary noise on Grover’s quantum search algorithm. Phys. Rev. A 67, 042301 (2003)

    ADS  Google Scholar 

  48. Reitzner, D., Hillery, M.: Grover search under localized dephasing. Phys. Rev. A 99, 012339 (2019)

    ADS  Google Scholar 

  49. Mandal, S.P., Ghoshal, A., Srivastava, C., Sen, U.: Invariance of success probability in Grover’s quantum search under local noise with memory. Phys. Rev. A 107, 022427 (2023)

    ADS  MathSciNet  Google Scholar 

  50. Pablo-Norman, B., Ruiz-Altaba, M.: Noise in Grover’s quantum search algorithm. Phys. Rev. A 61, 012301 (1999)

    ADS  Google Scholar 

  51. Long, G.L., Li, Y.S., Zhang, W.L., Tu, C.C.: Dominant gate imperfection in Grover’s quantum search algorithm. Phys. Rev. A 61, 042305 (2000)

    ADS  Google Scholar 

  52. Azuma, H.: Decoherence in Grover’s quantum algorithm: perturbative approach. Phys. Rev. A 65, 042311 (2002)

    ADS  Google Scholar 

  53. Rastegin, A.E., Shemet, A.M.: Quantum search degeneration under amplitude noise in queries to the oracle. Quant. Inf. Process. 21, 158 (2022)

    ADS  MathSciNet  Google Scholar 

  54. Pan, M., Xiong, T., Zhen, S.: Performance of Grover’s search algorithm with diagonalizable collective noises. Quant. Inf. Process. 22, 238 (2023)

    ADS  MathSciNet  Google Scholar 

  55. Shenvi, N., Brown, K.R., Whaley, K.B.: Effects of a random noisy oracle in search algorithm complexity. Phys. Rev. A 68, 052313 (2003)

    ADS  Google Scholar 

  56. Gawron, P., Klamka, J., Winiarczyk, R.: Noise effects in the quantum search algorithm from the viewpoint of computational complexity. Int. J. Appl. Math. Comput. Sci. 22, 493 (2012)

    MathSciNet  Google Scholar 

  57. Cohn, I., De Oliveira, A.L.F., Buksman, E., De Lacalle, J.G.L.: Grover’s search with local and total depolarizing channel errors: complexity analysis. Int. J. Quantum. Inform. 14, 1650009 (2016)

    ADS  MathSciNet  Google Scholar 

  58. Gebhart, V., Pezzè, L., Smerzi, A.: Quantifying computational advantage of Grover’s algorithm with the trace speed. Sci. Rep. 11, 1288 (2021)

    ADS  Google Scholar 

  59. Pokharel, B., Lidar, D.A.: Demonstration of algorithmic quantum speedup. Phys. Rev. Lett. 130, 210602 (2023)

    ADS  MathSciNet  Google Scholar 

  60. Fang, Y., Kaszlikowski, D., Chin, C., Tay, K., Kwek, L.C., Oh, C.H.: Entanglement in the Grover search algorithm. Phys. Lett. A 345, 265 (2005)

    ADS  Google Scholar 

  61. Shapira, D., Shimoni, Y., Biham, O.: Groverian measure of entanglement for mixed states. Phys. Rev. A 73, 044301 (2006)

    ADS  MathSciNet  Google Scholar 

  62. Shimoni, Y., Biham, O.: Groverian entanglement measure of pure quantum states with arbitrary partitions. Phys. Rev. A 75, 022308 (2007)

    ADS  Google Scholar 

  63. Rungta, P.: The quadratic speedup in Grover’s search algorithm from the entanglement perspective. Phys. Lett. A 373, 2652 (2009)

    ADS  Google Scholar 

  64. Cui, J., Fan, H.: Correlations in the Grover search. J. Phys. A Math. Theor. 43, 045305 (2010)

    ADS  MathSciNet  Google Scholar 

  65. Batle, J., Raymond Ooi, C.H., Farouk, A., Alkhambashi, M.S., Abdalla, S.: Global versus local quantum correlations in the Grover search algorithm. Quant. Inf. Process. 15, 833 (2016)

    ADS  MathSciNet  Google Scholar 

  66. Matera, J.M., Egloff, D., Killoran, N., Plenio, M.B.: Coherent control of quantum systems as a resource theory. Quantum Sci. Technol. 1, 01LT01 (2016)

  67. Pan, M., Qiu, D., Zheng, S.: Global multipartite entanglement dynamics in Grover’s search algorithm. Quant. Inf. Process. 16, 211 (2017)

    ADS  MathSciNet  Google Scholar 

  68. Gory, D.G., Fahmy, A.F., Havel, T.F.: Ensemble quantum computing by nuclear magnetic resonance spectroscopy. Proc. Natl. Acad. Sci. USA 94, 1634 (1997)

    ADS  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China, Grant No. 12005104.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China

    Yuan Sun

Authors
  1. Yuan Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Y.S. wrote the main manuscript text and prepared all the figures.

Corresponding author

Correspondence to Yuan Sun.

Ethics declarations

Conflict of interest

The authors declare no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sun, Y. Decoherence in Grover search algorithm. Quantum Inf Process 23, 183 (2024). https://doi.org/10.1007/s11128-024-04399-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-024-04399-6

Keywords

Search

Navigation

pFad - Phonifier reborn

Pfad - The Proxy pFad of © 2024 Garber Painting. All rights reserved.

Note: This service is not intended for secure transactions such as banking, social media, email, or purchasing. Use at your own risk. We assume no liability whatsoever for broken pages.


Alternative Proxies:

Alternative Proxy

pFad Proxy

pFad v3 Proxy

pFad v4 Proxy