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Experimental realization of non-Abelian non-adiabatic geometric gates

Nature volume 496, pages 482–485 (2013)Cite this article

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Abstract

The geometric aspects of quantum mechanics are emphasized most prominently by the concept of geometric phases, which are acquired whenever a quantum system evolves along a path in Hilbert space, that is, the space of quantum states of the system. The geometric phase is determined only by the shape of this path1,2,3 and is, in its simplest form, a real number. However, if the system has degenerate energy levels, then matrix-valued geometric state transformations, known as non-Abelian holonomies—the effect of which depends on the order of two consecutive paths—can be obtained4. They are important, for example, for the creation of synthetic gauge fields in cold atomic gases5 or the description of non-Abelian anyon statistics6,7. Moreover, there are proposals8,9 to exploit non-Abelian holonomic gates for the purposes of noise-resilient quantum computation. In contrast to Abelian geometric operations10, non-Abelian ones have been observed only in nuclear quadrupole resonance experiments with a large number of spins, and without full characterization of the geometric process and its non-commutative nature11,12. Here we realize non-Abelian non-adiabatic holonomic quantum operations13,14 on a single, superconducting, artificial three-level atom15 by applying a well-controlled, two-tone microwave drive. Using quantum process tomography, we determine fidelities of the resulting non-commuting gates that exceed 95 per cent. We show that two different quantum gates, originating from two distinct paths in Hilbert space, yield non-equivalent transformations when applied in different orders. This provides evidence for the non-Abelian character of the implemented holonomic quantum operations. In combination with a non-trivial two-quantum-bit gate, our method suggests a way to universal holonomic quantum computing.

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Figure 1: Geometric gate operation on a three-level system.
Figure 2: Transmon qubit in a cavity resonator.
Figure 3: Process tomography of holonomic gates.
Figure 4: Non-commutativity of holonomic gates.

References

  1. Pancharatnam, S. Generalized theory of interference and its applications. Proc. Indiana Acad. Sci. A44, 247–262 (1956)

    MathSciNet  Google Scholar 

  2. Berry, M. V. Quantal phase-factors accompanying adiabatic changes. Proc. R. Soc. Lond. A 392, 45–57 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  3. Aharonov, Y. & Anandan, J. Phase-change during a cyclic quantum evolution. Phys. Rev. Lett. 58, 1593–1596 (1987)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  4. Wilczek, F. & Zee, A. Appearance of gauge structure in simple dynamical systems. Phys. Rev. Lett. 52, 2111–2114 (1984)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  5. Dalibard, J., Gerbier, F., Juzeliūnas, G. & Öhberg, P. Artificial gauge potentials for neutral atoms. Rev. Mod. Phys. 83, 1523–1543 (2011)

    Article  ADS  CAS  Google Scholar 

  6. Nayak, C., Simon, S. H., Stern, A., Freedman, M. & Das Sarma, S. Non-Abelian anyons and topological quantum computation. Rev. Mod. Phys. 80, 1083–1159 (2008)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  7. Pachos, J. K. Introduction to Topological Quantum Computation (Cambridge Univ. Press, 2012)

    Book  Google Scholar 

  8. Zanardi, P. & Rasetti, M. Holonomic quantum computation. Phys. Lett. A 264, 94–99 (1999)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  9. Duan, L. M., Cirac, J. I. & Zoller, P. Geometric manipulation of trapped ions for quantum computation. Science 292, 1695–1697 (2001)

    Article  ADS  CAS  Google Scholar 

  10. Berry, M. Geometric phase memories. Nature Phys. 6, 148–150 (2010)

    Article  ADS  CAS  Google Scholar 

  11. Tycko, R. Adiabatic rotational splittings and Berry’s phase in nuclear-quadrupole resonance. Phys. Rev. Lett. 58, 2281–2284 (1987)

    Article  ADS  CAS  Google Scholar 

  12. Zwanziger, J. W., Koenig, M. & Pines, A. Non-abelian effects in a quadrupole system rotating around two axes. Phys. Rev. A 42, 3107–3110 (1990)

    Article  ADS  CAS  Google Scholar 

  13. Anandan, J. Non-adiabatic non-abelian geometric phase. Phys. Lett. A 133, 171–175 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  14. Sjöqvist, E. et al. Non-adiabatic holonomic quantum computation. N. J. Phys. 14, 103035 (2012)

    Article  MathSciNet  Google Scholar 

  15. Koch, J. et al. Charge-insensitive qubit design derived from the Cooper pair box. Phys. Rev. A 76, 042319 (2007)

    Article  ADS  Google Scholar 

  16. Sjöqvist, E. A new phase in quantum computation. Physics 1, 35 (2008)

    Article  Google Scholar 

  17. Leek, P. J. et al. Observation of Berry’s phase in a solid-state qubit. Science 318, 1889–1892 (2007)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  18. Möttönen, M., Vartiainen, J. J. & Pekola, J. P. Experimental determination of the Berry phase in a superconducting charge pump. Phys. Rev. Lett. 100, 177201 (2008)

    Article  ADS  Google Scholar 

  19. Filipp, S. et al. Experimental demonstration of the stability of Berry’s phase for a spin-1/2 particle. Phys. Rev. Lett. 102, 030404 (2009)

    Article  ADS  CAS  Google Scholar 

  20. Wu, H. et al. Geometric phase gates with adiabatic control in electron spin resonance. Phys. Rev. A 87, 032326 (2013)

    Article  ADS  Google Scholar 

  21. Faoro, L., Siewert, J. & Fazio, R. Non-abelian holonomies, charge pumping, and quantum computation with Josephson junctions. Phys. Rev. Lett. 90, 028301 (2003)

    Article  ADS  Google Scholar 

  22. Kamleitner, I., Solinas, P., Müller, C., Shnirman, A. & Möttönen, M. Geometric quantum gates with superconducting qubits. Phys. Rev. B 83, 214518 (2011)

    Article  ADS  Google Scholar 

  23. Solinas, P., Zanardi, P., Zanghi, N. & Rossi, F. Holonomic quantum gates: a semiconductor-based implementation. Phys. Rev. A 67, 062315 (2003)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  25. Paik, H. et al. Observation of high coherence in Josephson junction qubits measured in a three-dimensional circuit QED architecture. Phys. Rev. Lett. 107, 240501 (2011)

    Article  ADS  Google Scholar 

  26. Bianchetti, R. et al. Dynamics of dispersive single-qubit readout in circuit quantum electrodynamics. Phys. Rev. A 80, 043840 (2009)

    Article  ADS  Google Scholar 

  27. Motzoi, F., Gambetta, J. M., Rebentrost, P. & Wilhelm, F. K. Simple pulses for elimination of leakage in weakly nonlinear qubits. Phys. Rev. Lett. 103, 110501 (2009)

    Article  ADS  CAS  Google Scholar 

  28. Pechal, M. et al. Geometric phase and nonadiabatic effects in an electronic harmonic oscillator. Phys. Rev. Lett. 108, 170401 (2012)

    Article  ADS  CAS  Google Scholar 

  29. Johansson, M. et al. Robustness of non-adiabatic holonomic gates. Phys. Rev. A 86, 062322 (2012)

    Article  ADS  Google Scholar 

  30. Ježek, M., Fiurášek, J. & Hradil, Z. Quantum inference of states and processes. Phys. Rev. A 68, 012305 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  31. Bianchetti, R. et al. Control and tomography of a three level superconducting artificial atom. Phys. Rev. Lett. 105, 223601 (2010)

    Article  ADS  CAS  Google Scholar 

Download references

Acknowledgements

We are grateful to E. Sjöqvist and members of the EU-funded project GEOMDISS for discussions. We acknowledge feedback on the manuscript from P. Zanardi and I. Cirac. This work is supported financially by GEOMDISS, the Swiss National Science Foundation and ETH Zurich.

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Author notes

  1. J. M. Fink & K. Juliusson

    Present address: Present addresses: Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA (J.M.F.); Quantronics Group, Service de Physique de l’Etat Condensé (CNRS, URA 2464), IRAMIS, CEA-Saclay, 91191 Gif-sur-Yvette, France (K.J.).,

Authors and Affiliations

  1. Department of Physics, ETH Zürich, CH-8093, Zürich, Switzerland

    A. A. Abdumalikov Jr, J. M. Fink, K. Juliusson, M. Pechal, S. Berger, A. Wallraff & S. Filipp

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  1. A. A. Abdumalikov Jr
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  3. K. Juliusson
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  4. M. Pechal
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Contributions

A.A.A. and S.F. developed the idea for the experiment; A.A.A. performed the measurements and analysed the data; J.M.F. designed and fabricated the sample, J.M.F. and K.J. set up the experimental hardware and characterized the sample, M.P. and S.B. contributed to the experiment; A.A.A. and S.F. wrote the manuscript; A.W. and S.F. planned the project; all authors commented on the manuscript.

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Correspondence to A. A. Abdumalikov Jr.

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The authors declare no competing financial interests.

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Abdumalikov Jr, A., Fink, J., Juliusson, K. et al. Experimental realization of non-Abelian non-adiabatic geometric gates. Nature 496, 482–485 (2013). https://doi.org/10.1038/nature12010

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