We use cookies to improve your experience with our site. | More info.

Nature | Letter

日本語要約

Generation of three-qubit entangled states using superconducting phase qubits

Journal name:
Nature
Volume:
467,
Pages:
570–573
Date published:
DOI:
doi:10.1038/nature09418
Received
Accepted
Published online

Entanglement is one of the key resources required for quantum computation1, so the experimental creation and measurement of entangled states is of crucial importance for various physical implementations of quantum computers2. In superconducting devices3, two-qubit entangled states have been demonstrated and used to show violations of Bell’s inequality4 and to implement simple quantum algorithms5. Unlike the two-qubit case, where all maximally entangled two-qubit states are equivalent up to local changes of basis, three qubits can be entangled in two fundamentally different ways6. These are typified by the states |GHZright fence = (|000right fence + |111right fence)/ and |Wright fence = (|001right fence + |010right fence + |100right fence)/ . Here we demonstrate the operation of three coupled superconducting phase qubits7 and use them to create and measure |GHZright fence and |Wright fence states. The states are fully characterized using quantum state tomography8 and are shown to satisfy entanglement witnesses9, confirming that they are indeed examples of three-qubit entanglement and are not separable into mixtures of two-qubit entanglement.

View full text

Subject terms:

At a glance

Figures

View all figures
  1. Protocols for generating entangled states.
    Figure 1: Protocols for generating entangled states.

    a, Quantum circuit for generating |GHZright fence using CNOT gates. b, Quantum circuit for |GHZright fence that has been ‘recompiled’ to use iSWAP gates, which are directly generated by capacitive coupling in the phase qubit. These two circuits are not fully equivalent, but they both produce |GHZright fence when operating on the ground state as input. c, Circuit to generate |Wright fence using a single entangling step with simultaneous coupling between all three qubits. The entangling operation is turned on for a time tW = (4/9)tiSWAP, where tiSWAP is the time needed to complete an iSWAP gate between two qubits. In these quantum circuits, H represents the Hadamard gate, and X, Y and Z are rotations about the respective axes of the Bloch sphere by the subscript angles1. d, Capacitive coupling network to achieve symmetric coupling between all pairs of qubits (left), and simplified equivalent circuit using coupling to a central island (right). The complete network on the left requires six capacitors, and the coupling strength, g, is proportional to the qubit–qubit capacitance, CΔ. In the equivalent circuit on the right, the same coupling strength is attained by scaling the capacitors to Cc = 4CΔ, but now only four capacitors are required and the circuit can be easily laid out symmetrically on a chip.

  2. Device description and operation.
    Figure 2: Device description and operation.

    a, Schematic of coupled-qubit circuit. Each qubit is controlled individually by a flux bias line that sets the d.c. operating point, provides quasi-d.c. pulses for tuning the qubits in and out of resonance and provides a.c. (microwave) control signals for qubit rotations. In addition, each qubit is coupled to a superconducting quantum interference device (SQUID) for read-out of the qubit state. The qubits are capacitively coupled to the central island, which results in symmetric coupling between all pairs of qubits. b, Photomicrograph of the sample, fabricated with aluminium (light areas) on sapphire substrate (dark areas). The coupler is the cross-shaped structure in the centre, and the simplicity of this design is evident in the straightforward correspondence between the schematic and the completed device. The entire sample is mounted in a superconducting aluminium box and cooled to 25 mK in a dilution refrigerator.

  3. Generation of entangled states in the time domain.
    Figure 3: Generation of entangled states in the time domain.

    In each panel, the pulse sequence is shown on the left with time on the horizontal axis and qubit frequency on the vertical axis, and the measured state occupation probabilities (Pabc) are shown on the right. a, To characterize the three-qubit interaction, all qubits are initially detuned and qubit B is excited with a π-pulse. The qubits are then tuned into resonance to interact for some time, then detuned and measured. During the interaction, the excitation from qubit B (|010right fence) is swapped to qubits A and C (|100right fence and |001right fence), and then back again. Owing to the coupling symmetry, P100 and P001 are nearly equal throughout the entire sequence. At the crossing point where the three probabilities are equal, the system is in a |Wright fence state. b, The coupling is turned on until the crossing point is reached and then the qubits are detuned, leaving the system in a |Wright fence state. The small oscillations visible thereafter are caused by residual coupling due to the finite detuning. c, The |GHZright fence sequence is a translation of the circuit in Fig. 1b, with the iSWAP gates implemented by tuning the qubits pairwise into resonance for time tiSWAP = 40 ns and π/2 rotations implemented by 12-ns microwave pulses for a total length of 104 ns. At right, the probabilities are plotted versus time in each marked stage of the sequence. After creating the initial superposition (1), the two iSWAP gates change the phases of the various components of the state, with little effect on the populations (1–2, 2–3). The final rotations populate |000right fence and |111right fence while depopulating the other states. For an ideal |GHZright fence state, P000 and P111 should approach 50%, although in the experiment these levels are reduced owing to decoherence and errors, as discussed in the text. a.u., arbitrary units.

  4. Quantum state tomography of |GHZ[rang] and |W[rang].
    Figure 4: Quantum state tomography of |GHZright fence and |Wright fence.

    a, b, Real parts of the measured density matrices ρW (a) and ρGHZ (b). For both states, the theoretical density matrix has vanishing imaginary part, and the measured imaginary parts (not shown) are also found to be small, with |Im ρW| < 0.03 and |Im ρGHZ| < 0.10. c, d, Pauli set (generalized Stokes parameters) plotted for ρW (c) and ρGHZ (d). The bars show expectation values of combinations of Pauli operators on one, two and three qubits, with theory in grey and experiment overlaid in colour. The same state information is contained in both representations, but the Pauli sets clearly show the differences between |Wright fence-type and |GHZright fence-type entanglement. In addition to the three-qubit correlation terms, the |Wright fence state has two-qubit correlations because tracing out one qubit from a |Wright fence state leaves the others still partially entangled. The fidelity is FW = 0.78 ± 0.01. For |GHZright fence, the two-qubit correlations other than the trivial ZZ type are absent because tracing out one qubit leaves the others in a completely mixed state. The fidelity is FGHZ = 0.62 ± 0.01 and the state is also found to violate the Mermin–Bell inequality23, 24.

References

  1. Nielsen, M. A. & Chuang, I. L. Quantum Computation and Quantum Information (Cambridge Univ. Press, 2000)
  2. Ladd, T. D. et al. Quantum computers. Nature 464, 45–53 (2010)
  3. Clarke, J. & Wilhelm, F. K. Superconducting quantum bits. Nature 453, 1031–1042 (2008)
  4. Ansmann, M. et al. Violation of Bell’s inequality in Josephson phase qubits. Nature 461, 504–506 (2009)
  5. DiCarlo, L. et al. Demonstration of two-qubit algorithms with a superconducting quantum processor. Nature 460, 240–244 (2009)
  6. Dür, W., Vidal, G. & Cirac, J. I. Three qubits can be entangled in two inequivalent ways. Phys. Rev. A 62, 062314 (2000)
  7. McDermott, R. et al. Simultaneous state measurement of coupled Josephson phase qubits. Science 307, 1299–1302 (2005)
  8. Steffen, M. et al. Measurement of the entanglement of two superconducting qubits via state tomography. Science 313, 1423–1425 (2006)
  9. Acín, A., Bruss, D., Lewenstein, M. & Sanpera, A. Classification of mixed three-qubit states. Phys. Rev. Lett. 87, 040401 (2001)
  10. Barenco, A. et al. Elementary gates for quantum computation. Phys. Rev. A 52, 3457–3467 (1995)
  11. Shi, Y. Both Toffoli and controlled-NOT need little help to do universal quantum computation. Quantum Inf. Comput. 3, 84–92 (2003)
  12. Lanyon, B. P. et al. Simplifying quantum logic using higher-dimensional Hilbert spaces. Nature Phys. 5, 134–140 (2008)
  13. Monz, T. et al. Realization of the quantum Toffoli gate with trapped ions. Phys. Rev. Lett. 102, 040501 (2009)
  14. Cory, D. G. et al. Experimental quantum error correction. Phys. Rev. Lett. 81, 2152–2155 (1998)
  15. Galiautdinov, A. & Martinis, J. M. Maximally entangling tripartite protocols for Josephson phase qubits. Phys. Rev. A 78, 010305 (2008)
  16. DiCarlo, L. et al. Preparation and measurement of three-qubit entanglement in a superconducting circuit. Nature doi:10.1038/nature09416 (this issue).
  17. Wei, L. F., Liu, Y. X. & Nori, F. Generation and control of Greenberger-Horne-Zeilinger entanglement in superconducting circuits. Phys. Rev. Lett. 96, 246803 (2006)
  18. Matsuo, S. et al. Generation of macroscopic entangled states in coupled superconducting phase qubits. J. Phys. Soc. Jpn 76, 054802 (2007)
  19. Schuch, N. & Siewert, J. Natural two-qubit gate for quantum computation using the XY interaction. Phys. Rev. A 67, 032301 (2003)
  20. Lucero, E. et al. High-fidelity gates in a single Josephson qubit. Phys. Rev. Lett. 100, 247001 (2008)
  21. Neeley, M. et al. Emulation of a quantum spin with a superconducting phase qudit. Science 325, 722–725 (2009)
  22. Hofheinz, M. et al. Synthesizing arbitrary quantum states in a superconducting resonator. Nature 459, 546–549 (2009)
  23. Mermin, N. D. Extreme quantum entanglement in a superposition of macroscopically distinct states. Phys. Rev. Lett. 65, 1838–1840 (1990)
  24. Pan, J.-W., Bouwmeester, D., Daniell, M., Weinfurter, H. & Zeilinger, A. Experimental test of quantum nonlocality in three-photon Greenberger–Horne–Zeilinger entanglement. Nature 403, 515–519 (2000)

Download references

Author information

Affiliations

  1. Department of Physics, University of California, Santa Barbara, California 93106, USA

    • Matthew Neeley,
    • Radoslaw C. Bialczak,
    • M. Lenander,
    • E. Lucero,
    • Matteo Mariantoni,
    • A. D. O’Connell,
    • D. Sank,
    • H. Wang,
    • M. Weides,
    • J. Wenner,
    • Y. Yin,
    • T. Yamamoto,
    • A. N. Cleland &
    • John M. Martinis

  2. Green Innovation Research Laboratories, NEC Corporation, Tsukuba, Ibaraki 305-8501, Japan

    • T. Yamamoto

Contributions

M.N. fabricated the sample, performed the experiments and analysed the data. J.M.M. and E.L. designed the custom electronics. H.W. and T.Y. contributed to software infrastructure. All authors contributed to the fabrication process, qubit design, experimental set-up and manuscript preparation.

Competing financial interests

The authors declare no competing financial interests.

Corresponding author

Correspondence to:

Author details

Supplementary information

PDF files

  1. Supplementary Information (276K)

    This file contains Supplementary Methods and Discussion describing experimental methods and analysis procedures, Supplementary Tables 1-2, Supplementary Figure 1 with legend and additional references.

Additional data