Proposal of realizing superadiabatic geometric quantum computation in decoherence-free subspaces


We propose a practical scheme to implement universal superadiabatic geometric quantum gates in decoherence-free subspaces in the trapped-ions system. The logical qubit is only encoded by two neighboring physical qubits, which is the minimal resource for the decoherence-free subspace encoding. Different from the nonadiabatic control in decoherence-free subspace (Liang et al. in Phys Rev A 89:062312, 2014), a new Hamiltonian to implement universal effective interaction between logical qubits is proposed in the scheme. The proposed gates are numerically demonstrated to be robust against both systematic errors and collective dephasing noises, which combine the advantages of superadiabatic geometric quantum control and decoherence-free subspace. Since the Hamiltonian we use relies solely on two-body interactions, our scheme would be promising to be realized experimentally in trapped-ions systems.

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

Fig. 1
Fig. 2


  1. 1.

    Berry, M.V.: Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. A 392, 45 (1984)

    MathSciNet  MATH  ADS  Google Scholar 

  2. 2.

    Aharonov, Y., Anandan, J.: Phase change during a cyclic quantum evolution. Phys. Rev. Lett. 58, 1593 (1987)

    MathSciNet  ADS  Google Scholar 

  3. 3.

    De Chiara, G., Palma, G.M.: Berry phase for a spin 1/2 particle in a classical fluctuating field. Phys. Rev. Lett. 91, 090404 (2003)

    Google Scholar 

  4. 4.

    Solinas, P., Zanadi, P., Zanghì, N.: Robustness of non-Abelian holonomic quantum gates against parametric noise. Phys. Rev. A 70, 042316 (2004)

    MathSciNet  MATH  ADS  Google Scholar 

  5. 5.

    Solinas, P., Sassetti, M., Truini, P., Zanghì, N.: On the stability of quantum holonomic gates. New J. Phys. 14, 093006 (2012)

    MathSciNet  ADS  Google Scholar 

  6. 6.

    Zhu, S.L., Zanardi, P.: Geometric quantum gates that are robust against stochastic control errors. Phys. Rev. A 72, 020301(R) (2005)

    ADS  Google Scholar 

  7. 7.

    Liang, Z.T., Yue, X.X., Lv, Q.X., Du, Y.X., Huang, W., Yan, H., Zhu, S.L.: Proposal for implementing universal superadiabatic geometric quantum gates in nitrogen-vacancy centers. Phys. Rev. A 93, 040305(R) (2016)

    ADS  Google Scholar 

  8. 8.

    Berger, S., Pechal, M., Abdumalikov, A.A., Eichler, C., Steffen, L., Fedorov, A., Wallraff, A., Filipp, S.: Exploring the effect of noise on the Berry phase. Phys. Rev. A 87, 060303(R) (2013)

    ADS  Google Scholar 

  9. 9.

    Yale, C.G., Joseph Heremans, F., Zhou, B.B., Auer, A., Burkard, G., Awschalom, D.D.: Optical manipulation of the Berry phase in a solid-state spin qubit. Nat. Photon. 10, 184–189 (2016)

    ADS  Google Scholar 

  10. 10.

    Sjöqvist, E.: Trend: a new phase in quantum computation. Physics 1, 35 (2008)

    Google Scholar 

  11. 11.

    Sjöqvist, E.: Geometric phases in quantum information. Int. J. Quantum Chem. 115, 1311 (2015)

    Google Scholar 

  12. 12.

    Tan, X., Zhang, D.W., Zhang, Z., Yu, Y., Han, S., Zhu, S.L.: Demonstration of geometric Landau–Zener interferometry in a superconducting qubit. Phys. Rev. Lett 112, 027001 (2014)

    ADS  Google Scholar 

  13. 13.

    Zanardi, P., Rasetti, M.: Holonomic quantum computation. Phys. Lett. A 264, 94 (1999)

    MathSciNet  MATH  ADS  Google Scholar 

  14. 14.

    Pachos, J., Zanardi, P., Rasetti, M.: Non-Abelian Berry connections for quantum computation. Phys. Rev. A 61, 010305(R) (1999)

    MathSciNet  Google Scholar 

  15. 15.

    Duan, L.M., Cirac, J.I., Zoller, P.: Geometric manipulation of trapped ions for quantum computation. Science 292, 1695 (2001)

    ADS  Google Scholar 

  16. 16.

    Wang, X.B., Keiji, M.: Nonadiabatic conditional geometric phase shift with NMR. Phys. Rev. Lett. 87, 097901 (2001)

    ADS  Google Scholar 

  17. 17.

    Zhu, S.L., Wang, Z.D.: Implementation of universal quantum gates based on nonadiabatic geometric phases. Phys. Rev. Lett. 89, 097902 (2002)

    ADS  Google Scholar 

  18. 18.

    Zhu, S.L., Wang, Z.D.: Geometric phase shift in quantum computation using superconducting nanocircuits: nonadiabatic effects. Phys. Rev. A 66, 042322 (2002)

    ADS  Google Scholar 

  19. 19.

    Zhu, S.L., Wang, Z.D.: Universal quantum gates based on a pair of orthogonal cyclic states: application to NMR systems. Phys. Rev. A 67, 022319 (2003)

    ADS  Google Scholar 

  20. 20.

    Zhang, X.D., Zhu, S.L., Hu, L., Wang, Z.D.: Nonadiabatic geometric quantum computation using a single-loop scenario. Phys. Rev. A. 71, 014302 (2005)

    ADS  Google Scholar 

  21. 21.

    Zhu, S.L., Wang, Z.D.: Unconventional geometric quantum computation. Phys. Rev. Lett. 91, 187902 (2003)

    ADS  Google Scholar 

  22. 22.

    Zhu, S.L., Wang, Z.D., Zanardi, P.: Geometric quantum computation and multiqubit entanglement with superconducting qubits inside a cavity. Phys. Rev. Lett. 94, 100502 (2005)

    MathSciNet  ADS  Google Scholar 

  23. 23.

    Sjöqvist, E., Tong, D.M., Andersson, L.M.A.L.M., Hessmo, B., Johansson, M., Singh, K.: Non-adiabatic holonomic quantum computation. New J. Phys. 14, 103035 (2012)

    MathSciNet  ADS  Google Scholar 

  24. 24.

    Liang, Z.T., Du, Y.X., Huang, W., Xue, Z.Y., Yan, H.: Nonadiabatic holonomic quantum computation in decoherence-free subspaces with trapped ions. Phys. Rev. A 89, 062312 (2014)

    ADS  Google Scholar 

  25. 25.

    Xue, Z.Y., Zhou, J., Wang, Z.D.: Universal holonomic quantum gates in decoherence-free subspace on superconducting circuits. Phys. Rev. A 92, 022320 (2015)

    ADS  Google Scholar 

  26. 26.

    Xue, Z.-Y., Zhou, J., Hu, Y.: Nonadiabatic holonomic quantum computation with all-resonant control. Phys. Rev. A 94, 022331 (2016)

    ADS  Google Scholar 

  27. 27.

    Xu, G.F., Zhang, J., Tong, D.M., Sjöqvist, E., Kwek, L.C.: Nonadiabatic holonomic quantum computation in decoherence-free subspaces. Phys. Rev. Lett. 109, 170501 (2012)

    ADS  Google Scholar 

  28. 28.

    Abdumalikov, A.A., Fink, J.M., Juliusson, K., Pechal, M., Berger, S., Wallraff, A., Filipp, S.: Experimental realization of non-Abelian non-adiabatic geometric gates. Nature 496, 482 (2013)

    ADS  Google Scholar 

  29. 29.

    Feng, G., Xu, G., Long, G.: Experimental realization of nonadiabatic holonomic quantum computation. Phys. Rev. Lett. 110, 190501 (2013)

    ADS  Google Scholar 

  30. 30.

    Arroyo-Camejo, S., Lazariev, A., Hell, S.W., Balasubramanian, G.: Room temperature high-fidelity holonomic single-qubit gate on a solid-state spin. Nat. Commun. 5, 4870 (2014)

    ADS  Google Scholar 

  31. 31.

    Mousolou, V.A., Canall, C.M., Sjöqvist, E.: Universal non-adiabatic holonomic gates in quantum dots and single-molecule magnets. New J. Phys 16, 013029 (2014)

    Google Scholar 

  32. 32.

    Zu, C., Wang, W.B., He, L., Zhang, W.G., Dai, C.Y., Wang, F., Duan, L.M.: Experimental realization of universal geometric quantum gates with solid-state spins. Nature 514, 72 (2014)

    ADS  Google Scholar 

  33. 33.

    Wu, H., Gauger, E.M., George, R.E., Möttönen, M., Riemann, H., Abrosimov, N.V., Becker, P., Pohl, H.J., Itoh, K.M., Thewalt, M.L.W., Morton, J.J.L.: Geometric phase gates with adiabatic control in electron spin resonance. Phys. Rev. A 87, 032326 (2013)

    ADS  Google Scholar 

  34. 34.

    Zheng, S.B., Yang, C.P., Nori, F.: Comparison of the sensitivity to systematic errors between nonadiabatic non-Abelian geometric gates and their dynamical counterparts. Phys. Rev. A 93, 032313 (2016)

    ADS  Google Scholar 

  35. 35.

    Zhang, J., Kyaw, T.H., Tong, D.M., Sjöqvist, E., Kwek, L.C.: Fast non-Abelian geometric gates via transitionless quantum driving. Sci. Rep. 5, 18414 (2015)

    ADS  Google Scholar 

  36. 36.

    Song, X.K., Zang, H., Ai, Q., Qiu, J., Deng, F.G.: Shortcuts to adiabatic holonomic quantum computation in decoherence-free subspace with transitionless quantum driving algorithm. New J. Phys. 18, 023001 (2016)

    ADS  Google Scholar 

  37. 37.

    Du, Y.X., Liang, Z.T., Li, Y.C., Yue, X.X., Lv, Q.X., Huang, W., Chen, X., Yan, H., Zhu, S.L.: Experimental realization of stimulated Raman shortcut-to-adiabatic passage with cold atoms. Nat. Commun. 7, 12479 (2016)

    ADS  Google Scholar 

  38. 38.

    An, S., Lv, D., del Campo, A., Kim, K.: Shortcuts to adiabaticity by counterdiabatic driving for trapped-ion displacement in phase space. Nat. Commun. 7, 12999 (2016)

    ADS  Google Scholar 

  39. 39.

    Zhang, J., Shim, J.H., Niemeyer, I., Taniguchi, T., Teraji, T., Abe, H., Onoda, S., Yamamoto, T., Ohshima, T., Isoya, J., Suter, D.: Experimental implementation of assisted quantum adiabatic passage in a single spin. Phys. Rev. Lett. 110, 240501 (2013)

    ADS  Google Scholar 

  40. 40.

    Bason, M.G., Viteau, M., Malossi, N., Huillery, P., Arimondo, E., Ciampini, D., Fazio, R., Giovannetti, V., Mannella, R., Morsch, O.: High-fidelity quantum driving. Nat. Phys. 8, 147 (2012)

    Google Scholar 

  41. 41.

    Zhou, B.B., Baksic, A., Ribeiro, H., Yale, C.G., Heremans, F.J., Jerger, P.C., Auer, A., Burkard, G., Clerk, A.A., Awschalom, D.D.: Accelerated quantum control using superadiabatic dynamics in a solid-state lambda system. Nat. Phys. 13, 330–334 (2017)

    Google Scholar 

  42. 42.

    Suter, D., Álvarez, G.A.: Colloquium: protecting quantum information against environmental noise. Rev. Mod. Phys. 88, 041001 (2016)

    MathSciNet  ADS  Google Scholar 

  43. 43.

    Wu, L.A., Zanardi, P., Lidar, D.A.: Holonomic quantum computation in decoherence-free subspaces. Phys. Rev. Lett. 95, 130501 (2005)

    MathSciNet  ADS  Google Scholar 

  44. 44.

    Cen, L.X., Wang, Z.D., Wang, S.J.: Scalable quantum computation in decoherence-free subspaces with trapped ions. Phys. Rev. A 74, 032321 (2006)

    ADS  Google Scholar 

  45. 45.

    Zhang, X.D., Zhang, Q.H., Wang, Z.D.: Physical implementation of holonomic quantum computation in decoherence-free subspaces with trapped ions. Phys. Rev. A 74, 034302 (2006)

    ADS  Google Scholar 

  46. 46.

    Pachos, J.K., Beige, A.: Decoherence-free dynamical and geometrical entangling phase gates. Phys. Rev. A 69, 033817 (2004)

    ADS  Google Scholar 

  47. 47.

    Feng, X.L., Wu, C.F., Sun, H., Oh, C.H.: Geometric entangling gates in decoherence-free subspaces with minimal requirements. Phys. Rev. Lett. 103, 200501 (2009)

    ADS  Google Scholar 

  48. 48.

    Xue, Z.-Y., Zhu, S.-L., Wang, Z.D.: Quantum computation in a decoherence-free subspace with superconducting devices. Eur. Phys. J. D 55, 223 (2009)

    ADS  Google Scholar 

  49. 49.

    Sørensen, A., Mølmer, K.: Quantum computation with ions in thermal motion. Phys. Rev. Lett. 82, 1971 (1999)

    ADS  Google Scholar 

  50. 50.

    Sørensen, A., Mølmer, K.: Entanglement and quantum computation with ions in thermal motion. Phys. Rev. A 62, 022311 (2000)

    ADS  Google Scholar 

  51. 51.

    Zhu, S.L., Monroe, C., Duan, L.M.: Trapped Ion quantum computation with transverse phonon modes. Phys. Rev. Lett. 97, 050505 (2006)

    ADS  Google Scholar 

  52. 52.

    Zhu, S.L., Monroe, C., Duan, L.M.: Arbitrary-speed quantum gates within large ion crystals through minimum control of laser beams. Europhys. Lett. 73, 485 (2006)

    ADS  Google Scholar 

  53. 53.

    Schmidt-Kaler, F., Häffner, H., Riebe, M., Gulde, S., Lancaster, G.P.T., Deuschle, T., Becher, C., Roos, C.F., Eschner, J., Blatt, R.: Realization of the Cirac–Zoller controlled-NOT quantum gate. Nature 422, 408 (2003)

    ADS  Google Scholar 

  54. 54.

    Riebe, M., Kim, K., Schindler, P., Monz, T., Schmidt, P.O., Körber, T.K., Hänsel, W., Häffner, H., Roos, C.F., Blatt, R.: Process tomography of ion trap quantum gates. Phys. Rev. Lett. 97, 220407 (2006)

    ADS  Google Scholar 

  55. 55.

    Häffner, H., Gulde, S., Riebe, M., Lancaster, G., Becher, C., Eschner, J., Schmidt-Kaler, F., Blatt, R.: Precision measurement and compensation of optical Stark shifts for an ion-trap quantum processor. Phys. Rev. Lett. 90, 143602 (2003)

    ADS  Google Scholar 

  56. 56.

    Dzialoshinski, L.: A thermodynamic theory of weak ferromagnetism of antiferromagnetics. J. Phys. Chem. Solids 4, 241 (1958)

    ADS  Google Scholar 

  57. 57.

    Moriya, T.: New mechanism of anisotropic superexchange interaction. Phys. Rev. Lett. 4, 228 (1960)

    ADS  Google Scholar 

  58. 58.

    Kim, K., Roos, C.F., Aolita, L., Häfner, H., Nebendahl, V., Blatt, R.: Geometric phase gate on an optical transition for ion trap quantum computation. Phys. Rev. A 77, 050303(R) (2008)

    ADS  Google Scholar 

  59. 59.

    Monz, T., Kim, K., Villar, A.S., Schindler, P., Chwalla, M., Riebe, M., Roos, C.F., Häffner, H., Hänsel, W., Hennrich, M., Blatt, R.: Realization of universal ion-trap quantum computation with decoherence-free qubits. Phys. Rev. Lett. 103, 200503 (2009)

    ADS  Google Scholar 

  60. 60.

    Berry, M.V.: Transitionless quantum driving. J. Phys. A Math. Theor. 42, 365303 (2009)

    MathSciNet  MATH  Google Scholar 

  61. 61.

    Lindblad, G.: On the generators of quantum dynamical semigroups. Commun. Math. Phys. 48, 119 (1976)

    MathSciNet  MATH  ADS  Google Scholar 

  62. 62.

    Solinas, P., Zanardi, P., Zanghì, N.: Robustness of non-Abelian holonomic quantum gates against parametric noise. Phys. Rev. A 70, 042316 (2004)

    MathSciNet  MATH  ADS  Google Scholar 

  63. 63.

    Zahedinejad, E., Ghosh, J., Sanders, B.C.: High-fidelity single-shot toffoli gate via quantum control. Phys. Rev. Lett. 114, 200502 (2015)

    ADS  Google Scholar 

  64. 64.

    Lin, Y., Gaebler, J.P., Reiter, F., Tan, T.R., Bowler, R., Wan, Y., Keith, A., Knill, E., Glancy, S., Coakley, K., Sørensen, A.S., Leibfried, D., Wineland, D.J.: Preparation of entangled states through Hilbert space engineering. Phys. Rev. Lett. 117, 140502 (2016)

    ADS  Google Scholar 

  65. 65.

    Gaebler, J.P., Tan, T.R., Lin, Y., Wan, Y., Bowler, R., Keith, A.C., Glancy, S., Coakley, K., Knill, E., Leibfried, D., Wineland, D.J.: High-fidelity universal gate set for \({ ^{9}Be}^{+}\) ion qubits. Phys. Rev. Lett. 117, 060505 (2016)

    ADS  Google Scholar 

  66. 66.

    Chen, Q., Yang, W.L., Feng, M.: Quantum gate operations in decoherence-free fashion with separate nitrogen-vacancy centers coupled to a whispering-gallery mode resonator. Eur. Phys. J. D 66, 238 (2012)

    ADS  Google Scholar 

Download references


We thank X. X. Yue, Z. Y. Xue and S. L. Zhu for their helpful discussions. This work was supported by National Natural Science Foundation of China (NSFC) (11704131, 11474107, 91636218, 61875060); National Key Research and Development Program of China (NKRDPC) (2016YFA0301803, 2016YFA0302800); and the Natural Science Foundation of Guangdong province (2016A030310462).

Author information



Corresponding author

Correspondence to Yan-Xiong Du.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Li, J., Du, Y., Lv, Q. et al. Proposal of realizing superadiabatic geometric quantum computation in decoherence-free subspaces. Quantum Inf Process 18, 17 (2019).

Download citation


  • Superadiabatic geometric quantum computation
  • Decoherence-free subspace
  • Minimal requirements
  • Trapped ions