Skip to main content
SpringerLink
Log in

Deterministic generations of quantum state with no more than six qubits

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

The ability to prepare arbitrary quantum state is the holy grail of quantum information technology. Previous schemes focus on circuit complexity using implicit decomposition schemes for global evolutions and are difficult in quantum experiments because the generation circuit can be completed for given coefficients each time. One protocol is firstly proposed in this paper in order to deterministically generate arbitrary four-qubit states with any coefficients. In order to complete this scheme with present physical techniques, we present an explicit quantum circuit with unknown coefficients of prepared states using elementary quantum gates. The key of our scheme is constructing the Cartan KAK decomposition of special transformations in \(SO(4)\) and \(SO(8)\). And then, this protocol is extended to arbitrary five-qubit states and six-qubit states.

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

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Paige, C.C., Wei, M.: History and generality of the CS decomposition. Linear Algebra Appl. 208(209), 303–326 (1994)

  2. Barenco, A., Bennett, C.H., Cleve, R., DiVincenzo, D.P., Margolus, N., Shor, P., Sleator, T., Smolin, J.A., Weinfurter, H.: Elementary gates for quantum computation. Phys. Rev. A 52, 3457 (1995)

    Article  ADS  Google Scholar 

  3. Vidal, G., Dawson, C.M.: Universal quantum circuit for two-qubit transformations with three controlled-NOT gates. Phys. Rev. A 69, 010301 (2004)

    Article  ADS  Google Scholar 

  4. Shende, V., Markov, I., Bullock, S.: Minimal universal two-qubit controlled-NOT-based circuits. Phys. Rev. A 69, 062321 (2004)

    Article  ADS  Google Scholar 

  5. Shende, V., Bullock, S.S., Markov, I.L.: Synthesis of quantum-logic circuits. IEEE Trans. Comput. AID Des. 26, 1000–1010 (2006)

    Article  Google Scholar 

  6. Nielsen, M.A., Dowling, M.R., Gu, M., Doherty, A.C.: Quantum computation as geometry. Science 311, 1133 (2006)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  7. Brennen, G.K., O’Leary, D.P., Bullock, S.S.: Criteria for exact qudit universality. Phys. Rev. A 71, 052318 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  8. Li, B., Yu, Z.-H., Fei, S.-M.: Geometry of quantum computation with qutrits. Sci. Rep. 3, 2594 (2013)

    ADS  Google Scholar 

  9. Luo, M.X., Wang, X.: Universal quantum computation with qudits. Sci. China Phys. Mech. Astron. 57, 1712–1717 (2014)

    Article  ADS  Google Scholar 

  10. Luo, M.-X., Chen, X.-B., Yang, Y.-X., Wang, X.: Geometry of quantum computation with qudits. Sci. Rep. 4, 4044 (2014)

    ADS  Google Scholar 

  11. Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26, 1484–1509 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  12. Bimbard, W., Jain, N., MacRae, A., Lvovsky, A.I.: Quantum-optical state engineering up to the two-photon level. Nat. Photonics 4, 243–247 (2010)

    Article  ADS  Google Scholar 

  13. Kiesel, N., Schmid, C., Weber, U., Ursin, R., Weinfurter, H.: Linear optics controlled-phase gate made simple. Phys. Rev. Lett. 95, 210505 (2005)

    Article  ADS  Google Scholar 

  14. Xiao, Y.F., Zou, X.B., Han, Z.F., Guo, G.C.: Quantum phase gate in an optical cavity with atomic cloud. Phys. Rev. A 74, 044303 (2006)

    Article  ADS  Google Scholar 

  15. Song, J., Xia, Y., Song, H.-S., Guo, J.-L., Nie, J.: Quantum computation and entangled-state generation through adiabatic evolution in two distant cavities. EPL 80, 60001 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  16. VanMeter, N.M., Lougovski, P., Uskov, D.B., Kieling, K., Eisert, J., Dowling, J.P.: General linear-optical quantum state generation scheme: applications to maximally path-entangled states. Phys. Rev. A 76, 063808 (2007)

    Article  ADS  Google Scholar 

  17. Hofheinz, M., Weig, E.M., Ansmann, M., Bialczak, R.C., Lucero, E., Neeley, M., O’Connell, A.D., Wang, H., Martinis, J.M., Cleland, A.N.: Generation of Fock states in a superconducting quantum circuit. Nature 454, 310–314 (2008)

    Article  ADS  Google Scholar 

  18. Lu, C.-Y., Zhou, X.-Q., Guhne, O., Gao, W.B., Zhang, J., Yuan, Z.S., Goebel, A., Yang, T., Pan, J.-W.: Experimental entanglement of six photons in graph states. Nat. Phys. 3, 91–95 (2007)

    Article  Google Scholar 

  19. Gao, W.-B., Lu, C.-Y., Yao, X.-C., Xu, P., Guhne, O., Goebel, A., Chen, Y.A., Peng, C.Z., Chen, Z.B., Pan, J.-W.: Experimental demonstration of a hyper-entangled ten-qubit Schrodinger cat state. Nat. Phys. 6, 331–335 (2010)

    Article  Google Scholar 

  20. Wagenknecht, C., Li, C.M., Reingruber, A., Bao, X.H., Goebel, A., Chen, Y.A., Zhang, Q., Chen, K., Pan, J.-W.: Experimental demonstration of a heralded entanglement source. Nat. Photonics 4, 549–552 (2010)

    Article  ADS  Google Scholar 

  21. Roos, C.F., Riebe, M., Hoffner, H., Honsel, W., Benhelm, J., Lancaster, G.P.T., Becher, C., Schmidt-Kaler, F., Blatt, R.: Control and measurement of three-qubit entangled states. Science 304, 1478–1480 (2004)

    Article  ADS  Google Scholar 

  22. Yu, C.S., Yi, X.X., Song, H.S., Mei, D.: Robust preparation of Greenberger-Horne-Zeilinger and W states of three distant atoms. Phys. Rev. A 75, 044301 (2007)

    Article  ADS  Google Scholar 

  23. DiCarlo, L., Reed, M.D., Sun, L., Johnson, B.R., Chow, J.M., Gambetta, J.M., Frunzio, L., Girvin, S.M., Devoret, M.H., Schoelkopf, R.J.: Preparation and measurement of three-qubit entanglement in a superconducting circuit. Nature 467, 574–578 (2010)

    Article  ADS  Google Scholar 

  24. Neeley, M., Bialczak, R.C., Lenander, M., Lucero, E., Mariantoni, M., O’Connell, A.D., Sank, D., Wang, H., Weides, M., Wenner, J., Yin, Y., Yamamoto, T., Cleland, A.N., Martinis, J.M.: Generation of three-qubit entangled states using superconducting phase qubits. Nature 467, 570–573 (2010)

    Article  ADS  Google Scholar 

  25. Plesch, M., Brukner, C.: Quantum state preparation with universal gate decompositions. Phys. Rev. A 83, 032302 (2011)

    Article  ADS  Google Scholar 

  26. Luo, M.-X., Chen, X.-B., Yang, Y.-X., Niu, X.-X.: Experiment architecture of joint remote state preparation. Quantum Inf. Proc. 11, 751–767 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  27. Cartan, E.: Sur certaines formes riemanniennes remarquables des geometriesa groupe fondamental simple. Ann. Sci. Ec. Norm. Super. 44, 345 (1927)

    MATH  MathSciNet  Google Scholar 

  28. Khaneja, N., Glaser, S.: Cartan decomposition of \(SU(n)\) and control of spin systems. Chem. Phys. 267, 11–23 (2001)

    Article  ADS  Google Scholar 

  29. Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore (1996)

  30. Luo, M.-X., Chen, X.-B., Ma, S.-Y., Niu, X.-X., Yang, Y.-X.: Joint remote preparation of an arbitrary three-qubit state. Opt. Commun. 83, 4796–4801 (2010)

    Article  ADS  Google Scholar 

  31. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  32. Cernoch, A., Soubusta, J., Bartukova, L., Dusek, M., Fiurasek, J.: Experimental realization of linear-optical partial swap gates. Phys. Rev. Lett. 100, 180501 (2008)

    Article  ADS  Google Scholar 

  33. Lanyon, B.P., Barbieri, M., Almeida, M.P., Jennewein, T., Ralph, T.C., Resch, K.J., Pryde, G.J., O’Brien, J.L., Gilchrist, A., White, A.G.: Simplifying quantum logic using higher-dimensional Hilbert spaces. Nat. Phys. 5, 134–140 (2009)

    Article  Google Scholar 

  34. Monz, T., Kim, K., Hensel, W., Riebe, M., Villar, A.S., Schindler, P., Chwalla, M., Hennrich, M., Blatt, R.: Realization of the quantum Toffoli gate with trapped Ions. Phys. Rev. Lett. 102, 040501 (2009)

    Article  ADS  Google Scholar 

  35. Fedorov, A., Steffen, L., Baur, M., da Silva, M.P., Wallraff, A.: Implementation of a Toffoli gate with superconducting circuits. Nature 481, 170–172 (2012)

    Article  ADS  Google Scholar 

  36. Zheng, S.B.: Simplified construction and physical realization of n-qubit controlled phase gates. Phys. Rev. A 86, 012326 (2012)

    Article  ADS  Google Scholar 

  37. Liu, J.-M., Feng, X.-L., Oh, C.H.: Remote preparation of arbitrary two- and three-qubit states. EPL 87, 30006 (2009)

    Article  ADS  Google Scholar 

  38. Zhan, Y.-B.: Deterministic remote preparation of arbitrary two- and three-qubit states. EPL 98, 40005 (2012)

    Article  ADS  Google Scholar 

  39. Möttöen, M., Vartiainen, J.J., Bergholm, V., Salomaa, M.M.: Quantum circuits for general multiqubit gates. Phys. Rev. Lett. 93, 130502 (2004)

    Article  Google Scholar 

  40. Saeedi, M., Arabzadeh, M., Zamani, M. S., Sedighi, M.: Block-based quantum-logic synthesis, arXiv:1011.2159 (2010)

  41. Möttöen, M., Vartiainen, J.J., Bergholm, V., Salomaa, M.M.: Transformation of quantum states using uniformly controlled rotations. Quant. Inf. Comput. 5, 467 (2005)

    Google Scholar 

  42. Vartiainen, J., Möttöen, M., Salomaa, M.M.: Efficient decomposition of quantum gates. Phys. Rev. Lett. 92, 177902 (2004)

    Article  ADS  Google Scholar 

  43. Wille, R., Saeedi, M., Drechsler, R.: Synthesis of Reversible Functions Beyond Gate Count and Quantum Cost. International Workshop on Logic Synthesis (IWLS), San Francisco (2009)

    Google Scholar 

  44. Bullock, S.S., O’Leary, D.P., Brennen, G.K.: Asymptotically optimal quantum circuits for \(d\)-level systems. Phys. Rev. Lett. 94, 230502 (2005)

    Article  ADS  Google Scholar 

  45. Hladký, B., Drobný, G., Bužk, V.: Quantum synthesis of arbitrary unitary operators. Phys. Rev. A 61, 022102 (2000)

    Article  ADS  Google Scholar 

  46. Grover, L.K.: Synthesis of quantum superpositions by quantum computation. Phys. Rev. Lett. 85, 1334 (2000)

    Article  ADS  Google Scholar 

  47. Zhang, J., Lu, Z., Shan, L., Deng, Z.: Synthesizing NMR analogs of Einstein-Podolsky-Rosen states using the generalized Grover’s algorithm. Phys. Rev. A 66, 044308 (2002)

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  49. Hayato, G., Kouichi, I.: Multiqubit controlled unitary gate by adiabatic passage with an optical cavity. Phys. Rev. A 70, 012305 (2004)

    Article  Google Scholar 

  50. Lin, X.-M., Zhou, Z.-W., Ye, M.-Y., Xiao, Y.-F., Guo, G.-C.: One-step implementation of a multiqubit controlled-phase-flip gate. Phys. Rev. A 73, 012323 (2006)

    Article  ADS  Google Scholar 

Download references

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Nos.61303039, 61201253), Fundamental Research Funds for the Central Universities (No.2682014CX095), Open Foundation of State key Laboratory of Networking and Switching Technology (Beijing University of Posts and Telecommunications) (No.SKLNST-2013-1-11), Science Foundation Ireland under the International Strategic Cooperation Award Grant Number SFI/13/ISCA/2845.

Author information

Authors and Affiliations

  1. Information Security and National Computing Grid Laboratory, Southwest Jiaotong University, Chengdu, 610031, China

    Ming-Xing Luo

  2. State key Laboratory of Networking and Switching Technology (Beijing University of Posts and Telecommunications), Beijing, 100876, China

    Ming-Xing Luo

  3. State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing, 100093, China

    Ming-Xing Luo

  4. School of Mathematics and Statistics, Henan University, Kaifeng, 475004, China

    Song-Ya Ma

  5. Institute of Computer Science, Sichuan University of Science Engineering, Zigong, 64300, China

    Yun Deng

  6. School of Electronic Engineering, Dublin City University, Dublin 9, Ireland

    Xiaojun Wang

Authors
  1. Ming-Xing Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Song-Ya Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yun Deng
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Xiaojun Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ming-Xing Luo.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Luo, MX., Ma, SY., Deng, Y. et al. Deterministic generations of quantum state with no more than six qubits. Quantum Inf Process 14, 901–920 (2015). https://doi.org/10.1007/s11128-014-0905-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11128-014-0905-9

Keywords

Search

Navigation