Skip to main content
SpringerLink
Log in

Can quantum entanglement detection schemes improve search?

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

Quantum computation, in particular Grover’s algorithm, has aroused a great deal of interest since it allows for a quadratic speed-up to be obtained in search procedures. Classical search procedures for an N element database require at most O(N) time complexity. Grover’s algorithm is able to find a solution with high probability in \({O(\sqrt{N})}\) time through an amplitude amplification scheme. In this work we draw elements from both classical and quantum computation to develop an alternative search proposal based on quantum entanglement detection schemes. In 2002, Horodecki and Ekert proposed an efficient method for direct detection of quantum entanglement. Our proposition to quantum search combines quantum entanglement detection alongside entanglement inducing operators. The quantum search algorithm relies on measuring a quantum superposition after having applied a unitary evolution. We deviate from the standard method by focusing on fine-tuning a unitary operator in order to infer the solution with certainty. Our proposal sacrifices space for speed and depends on the mathematical properties of linear positive maps Λ which have not been operationally characterized. Whether such a Λ can be easily determined remains an open question.

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

Similar content being viewed by others

References

  1. Barenco, A., Berthiaume, A., Deutsch, D., Ekert, A., Jozsa, R., Macchiavello, C.: Stabilization of quantum computations by symmetrization. SIAM J. Comput. 26(5),1541–1557 (1997) doi:10.1137/S0097539796302452. http://link.aip.org/link/?SMJ/26/1541/1

    Google Scholar 

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

    Article  MATH  Google Scholar 

  3. Bennett, C.H., Bernstein, E., Brassard, G., Vazirani, U.: Strengths and weaknesses of quantum computing (1997) http://www.citebase.org/abstract?id=oai:arXiv.org:quant-ph/9701001

  4. Cook, S.A.: The complexity of theorem-proving procedures. In: STOC ’71: Proceedings of the Third Annual ACM Symposium on Theory of Computing, pp. 151–158. ACM, New York, NY, USA (1971) http://doi.acm.org/10.1145/800157.805047

  5. Cormen T.H., Leiserson C.E., Rivest R.L., Stein C.: Introduction to Algorithms, 2/e. MIT Press, Cambridge (2001)

    Google Scholar 

  6. Deutsch D., Jozsa R.: Rapid solution of problems by quantum computation. R. Soc. Lond. Proc. Ser. A 439, 553–558 (1992)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  7. Einstein A., Podolsky B., Rosen N.: Can quantum-mechanical description of physical reality be considered complete?. Phys. Rev. 47(10), 777–780 (1935). doi:10.1103/Phys.Rev.47.777

    Article  MATH  ADS  Google Scholar 

  8. Gharibian, S.: Strong NP-Hardness of the Quantum Separability Problem. ArXiv e-prints (2008)

  9. Grover, L.K.: A fast quantum mechanical algorithm for database search. In: STOC ’96: Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, pp. 212–219. ACM, New York, NY, USA (1996) http://doi.acm.org/10.1145/237814.237866

  10. Grover, L.K., Radhakrishnan, J.: Is partial quantum search of a database any easier? (2004) http://www.citebase.org/abstract?id=oai:arXiv.org:quant-ph/0407122

  11. Gühne O., Tóth G.: Entanglement detection. Phys. Rep. 474, 1–75 (2009). doi:10.1016/j.physrep.2009.02.004

    Article  ADS  MathSciNet  Google Scholar 

  12. Gurvits, L.: Classical deterministic complexity of edmonds’ problem and quantum entanglement. In: STOC ’03: Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing, pp. 10–19. ACM, New York, NY, USA (2003) http://doi.acm.org/10.1145/780542.780545

  13. Horodecki, M., Horodecki, P., Horodecki, R.: Separability of mixed states:necessary and sufficient conditions. Phys. Lett. A 223(1–2),1–8 (1996) doi:10.1016/S0375-9601(96)00706-2. http://www.sciencedirect.com/science/article/B6TVM-3VSFHG4-1J/2/30233fc8e862b1e50e0d0a7e340f7859

    Google Scholar 

  14. Horodecki, M., Horodecki, P., Horodecki, R.: Separability of n-particle mixed states: necessary and sufficient conditions in terms of linear maps. Phys. Lett. A 283(1–2),1–7 (2001) doi:10.1016/S0375-9601(01)00142-6. http://www.sciencedirect.com/science/article/B6TVM-42YFC9G-1/2/9c9d0b6d096a6b59a89d0b03fe825977

  15. Horodecki P., Ekert A.: Method for direct detection of quantum entanglement. Phys. Rev. Lett. 89(12), 127,902 (2002). doi:10.1103/PhysRevLett.89.127902

    Article  MathSciNet  Google Scholar 

  16. Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. ArXiv Quantum Physics e-prints (2007)

  17. Ioannou, L.M.: Computational complexity of the quantum separability problem. ArXiv Quantum Physics e-prints (2006)

  18. Ioannou L.M., Travaglione B.C.: Quantum separability and entanglement detection via entanglement-witness search and global optimization. Phys. Rev. A 73(5), 052,314 (2006). doi:10.1103/PhysRevA.73.052314

    Article  Google Scholar 

  19. Kaye P.R., Laflamme R., Mosca M.: An Introduction to Quantum Computing. Oxford University Press, USA (2007)

    MATH  Google Scholar 

  20. Keyl M., Werner R.F.: Estimating the spectrum of a density operator. Phys. Rev. A 64(5), 052,311 (2001). doi:10.1103/PhysRevA.64.052311

    Article  MathSciNet  Google Scholar 

  21. Krammer, P.: Quantum entanglement—detection, classification, and quantification. Master’s thesis, University of Vienna, (2005)

  22. Lewenstein M., Kraus B., Cirac J.I., Horodecki P.: Optimization of entanglement witnesses. Phys. Rev. A 62(5), 052,310 (2000). doi:10.1103/PhysRevA.62.052310

    Article  Google Scholar 

  23. Lewenstein M., Kraus B., Horodecki P., Cirac J.I.: Characterization of separable states and entanglement witnesses. Phys. Rev. A 63(4), 044,304 (2001). doi:10.1103/PhysRevA.63.044304

    Article  MathSciNet  Google Scholar 

  24. von Neumann J.: Mathematische Grund lagen der Quantenmechanic. Springer, Berlin (1932)

    Google Scholar 

  25. Peres A.: Separability criterion for density matrices. Phys. Rev. Lett. 77(8), 1413–1415 (1996). doi:10.1103/PhysRevLett.77.1413

    Article  MATH  ADS  MathSciNet  Google Scholar 

  26. Schrödinger, E.: Die gegenwärtige situation in der quantenmechanik. Naturwissenschaften 23(807) (1935)

  27. Shor, P.:Algorithms for quantum computation: discrete logarithms and factoring. In: Proceedings 35th Annual Symposium on Foundations of Computer Science, pp. 124–134 (1994) doi:10.1109/SFCS.1994.365700

  28. Fiurášek J.: Structural physical approximations of unphysical maps and generalized quantum measurements. Phys. Rev. A 66(5), 052,315 (2002). doi:10.1103/PhysRevA.66.052315

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. GAIPS/INESC-ID, Department of Computer Science, Instituto Superior Técnico, Technical University of Lisbon, Avenida Professor Cavaco Silva, 2780-990, Porto Salvo, Portugal

    Luís Tarrataca & Andreas Wichert

Authors
  1. Luís Tarrataca
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andreas Wichert
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Luís Tarrataca.

Additional information

This work was supported by FCT (INESC-ID multiannual funding) through the PIDDAC Program funds and FCT grant DFRH–SFRH/BD/61846/2009.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tarrataca, L., Wichert, A. Can quantum entanglement detection schemes improve search?. Quantum Inf Process 11, 55–66 (2012). https://doi.org/10.1007/s11128-011-0231-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11128-011-0231-4

Keywords

Search

Navigation