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Implementation of a Toffoli gate with superconducting circuits

Nature volume 481, pages 170–172 (2012)Cite this article

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Abstract

The Toffoli gate is a three-quantum-bit (three-qubit) operation that inverts the state of a target qubit conditioned on the state of two control qubits. It makes universal reversible classical computation1 possible and, together with a Hadamard gate2, forms a universal set of gates in quantum computation. It is also a key element in quantum error correction schemes3,4,5,6,7. The Toffoli gate has been implemented in nuclear magnetic resonance3, linear optics8 and ion trap systems9. Experiments with superconducting qubits have also shown significant progress recently: two-qubit algorithms10 and two-qubit process tomography have been implemented11, three-qubit entangled states have been prepared12,13, first steps towards quantum teleportation have been taken14 and work on quantum computing architectures has been done15. Implementation of the Toffoli gate with only single- and two-qubit gates requires six controlled-NOT gates and ten single-qubit operations16, and has not been realized in any system owing to current limits on coherence. Here we implement a Toffoli gate with three superconducting transmon qubits coupled to a microwave resonator. By exploiting the third energy level of the transmon qubits, we have significantly reduced the number of elementary gates needed for the implementation of the Toffoli gate, relative to that required in theoretical proposals using only two-level systems. Using full process tomography and Monte Carlo process certification, we completely characterize the Toffoli gate acting on three independent qubits, measuring a fidelity of 68.5 ± 0.5 per cent. A similar approach15 to realizing characteristic features of a Toffoli-class gate has been demonstrated with two qubits and a resonator and achieved a limited characterization considering only the phase fidelity. Our results reinforce the potential of macroscopic superconducting qubits for the implementation of complex quantum operations with the possibility of quantum error correction17.

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Figure 1: Circuit diagram of the Toffoli gate.
Figure 2: Truth table of the Toffoli gate.
Figure 3: Process tomography of the Toffoli gate.

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References

  1. Toffoli, T. Reversible Computing (Lect. Notes Computer Sci. 85, Springer, 1980)

    Book  Google Scholar 

  2. Shi, Y. Both Toffoli and controlled-NOT need little help to do universal quantum computation. Quantum Inf. Comput. 3, 84–92 (2003)

    MathSciNet  MATH  Google Scholar 

  3. Cory, D. G. et al. Experimental quantum error correction. Phys. Rev. Lett. 81, 2152–2155 (1998)

    Article  ADS  CAS  Google Scholar 

  4. Knill, E., Laflamme, R., Martinez, R. & Negrevergne, C. Benchmarking quantum computers: the five-qubit error correcting code. Phys. Rev. Lett. 86, 5811–5814 (2001)

    Article  ADS  CAS  Google Scholar 

  5. Chiaverini, J. et al. Realization of quantum error correction. Nature 432, 602–605 (2004)

    Article  ADS  CAS  Google Scholar 

  6. Pittman, T. B., Jacobs, B. C. & Franson, J. D. Demonstration of quantum error correction using linear optics. Phys. Rev. A 71, 052332 (2005)

    Article  ADS  Google Scholar 

  7. Aoki, T. et al. Quantum error correction beyond qubits. Nature Phys. 5, 541–546 (2009)

    Article  ADS  CAS  Google Scholar 

  8. Lanyon, B. P. et al. Simplifying quantum logic using higher-dimensional Hilbert spaces. Nature Phys. 5, 134–140 (2009)

    Article  ADS  CAS  Google Scholar 

  9. Monz, T. et al. Realization of the quantum Toffoli gate with trapped ions. Phys. Rev. Lett. 102, 040501 (2009)

    Article  ADS  CAS  Google Scholar 

  10. DiCarlo, L. et al. Demonstration of two-qubit algorithms with a superconducting quantum processor. Nature 460, 240–244 (2009)

    Article  ADS  CAS  Google Scholar 

  11. Yamamoto, T. et al. Quantum process tomography of two-qubit controlled-z and controlled-not gates using superconducting phase qubits. Phys. Rev. B 82, 184515 (2010)

    Article  ADS  Google Scholar 

  12. DiCarlo, L. et al. Preparation and measurement of three-qubit entanglement in a superconducting circuit. Nature 467, 574–578 (2010)

    Article  ADS  CAS  Google Scholar 

  13. Neeley, M. et al. Generation of three-qubit entangled states using superconducting phase qubits. Nature 467, 570–573 (2010)

    Article  ADS  CAS  Google Scholar 

  14. Baur, M. et al. Benchmarking a teleportation protocol realized in superconducting circuits. Preprint at 〈http://arxiv.org/abs/1107.4774〉 (2011)

  15. Mariantoni, M. et al. Implementing the quantum Von Neumann architecture with superconducting circuits. Science 334, 61–65 (2011)

    Article  ADS  CAS  Google Scholar 

  16. Barenco, A. et al. Elementary gates for quantum computation. Phys. Rev. A 52, 3457–3467 (1995)

    Article  ADS  CAS  Google Scholar 

  17. Reed, M. D. et al. Realization of three-qubit quantum error correction with superconducting circuits. Preprint at 〈http://arxiv.org/abs/1109.4948〉 (2011)

  18. Filipp, S. et al. Two-qubit state tomography using a joint dispersive readout. Phys. Rev. Lett. 102, 200402 (2009)

    Article  ADS  CAS  Google Scholar 

  19. Majer, J. et al. Coupling superconducting qubits via a cavity bus. Nature 449, 443–447 (2007)

    Article  ADS  CAS  Google Scholar 

  20. Ralph, T. C., Resch, K. J. & Gilchrist, A. Efficient Toffoli gates using qudits. Phys. Rev. A 75, 022313 (2007)

    Article  ADS  Google Scholar 

  21. Borrelli, M., Mazzola, L., Paternostro, M. & Maniscalco, S. Simple trapped-ion architecture for high-fidelity Toffoli gates. Phys. Rev. A 84, 012314 (2011)

    Article  ADS  Google Scholar 

  22. Spörl, A. et al. Optimal control of coupled Josephson qubits. Phys. Rev. A 75, 012302 (2007)

    Article  ADS  Google Scholar 

  23. Stojanovic, V. M., Fedorov, A., Bruder, C. & Wallraff, A. Quantum-control approach to realizing a Toffoli gate in circuit QED. Preprint at 〈http://arxiv.org/abs/1108.3442〉 (2011)

  24. Gambetta, J. M., Motzoi, F., Merkel, S. T. & Wilhelm, F. K. Analytic control methods for high-fidelity unitary operations in a weakly nonlinear oscillator. Phys. Rev. A 83, 012308 (2011)

    Article  ADS  Google Scholar 

  25. 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 

  26. Strauch, F. W. et al. Quantum logic gates for coupled superconducting phase qubits. Phys. Rev. Lett. 91, 167005 (2003)

    Article  ADS  Google Scholar 

  27. Haack, G., Helmer, F., Mariantoni, M., Marquardt, F. & Solano, E. Resonant quantum gates in circuit quantum electrodynamics. Phys. Rev. B 82, 024514 (2010)

    Article  ADS  Google Scholar 

  28. Chuang, I. L. & Nielsen, M. A. Prescription for experimental determination of the dynamics of a quantum black box. J. Mod. Opt. 44, 2455–2467 (1997)

    Article  ADS  Google Scholar 

  29. da Silva, M. P., Landon-Cardinal, O. & Poulin, D. Practical characterization of quantum devices without tomography. Phys. Rev. Lett. 107, 210404 (2011)

    Article  ADS  Google Scholar 

  30. Flammia, S. T. & Liu, Y.-K. Direct fidelity estimation from few Pauli measurements. Phys. Rev. Lett. 106, 230501 (2011)

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  32. Bylander, J. et al. Noise spectroscopy through dynamical decoupling with a superconducting flux qubit. Nature Phys. 7, 565–570 (2011)

    Article  ADS  CAS  Google Scholar 

  33. Paik, H. et al. How coherent are Josephson junctions? Preprint at 〈http://arxiv.org/abs/1105.4652v1〉 (2011)

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Acknowledgements

We thank S. Filipp, A. Blais for useful discussions and K. Pakrouski for his contributions in early stages of the experimental work. This work was supported by the Swiss National Science Foundation, the EU IP SOLID and ETH Zurich.

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Authors and Affiliations

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

    A. Fedorov, L. Steffen, M. Baur & A. Wallraff

  2. Disruptive Information Processing Technologies Group, Raytheon BBN Technologies, 10 Moulton Street, Cambridge, Massachusetts 02138, USA,

    M. P. da Silva

  3. Département de Physique, Université de Sherbrooke, Sherbrooke, Québec J1K 2R1, Canada,

    M. P. da Silva

Authors
  1. A. Fedorov
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  2. L. Steffen
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  3. M. Baur
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  4. M. P. da Silva
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  5. A. Wallraff
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Contributions

A.F. developed the scheme to realize the Toffoli gate. L.S., A.F. and M.B. carried out the experiments and analysed the data. L.S. designed and fabricated the superconducting resonator. A.F. and M.B. designed and fabricated the qubits. M.P.d.S. provided general theoretical support and specific advice on using Monte Carlo process certification. A.F., L.S., M.B. and A.W. contributed to setting up and maintaining the experiment. A.F., A.W. and L.S. wrote the manuscript. All authors commented on the manuscript. A.W. supervised the project.

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Correspondence to A. Fedorov or A. Wallraff.

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The authors declare no competing financial interests.

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Fedorov, A., Steffen, L., Baur, M. et al. Implementation of a Toffoli gate with superconducting circuits. Nature 481, 170–172 (2012). https://doi.org/10.1038/nature10713

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