Research ArticlePHYSICS

Experimental benchmarking of quantum control in zero-field nuclear magnetic resonance

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Science Advances  15 Jun 2018:
Vol. 4, no. 6, eaar6327
DOI: 10.1126/sciadv.aar6327
  • Fig. 1 Zero-field NMR of 13C-formic acid.

    (A) Schematic molecular structure and zero-field nuclear spin energy levels of 13C-formic acid (1H-13COOH); single-shot zero-field NMR signal. The FWHM (full width at half maximum) obtained from a Lorentzian fit is 32 mHz. (B) Experimental setup for zero-field NMR spectroscopy, described in Materials and Methods. The NMR sample is contained in a 5-mm NMR tube and pneumatically shuttled between a 1.8-T prepolarizing magnet and the interior of a four-layer magnetic shield. A guiding field is applied in the z direction during the pneumatic shuttling. NMR signals are detected with an atomic magnetometer with a 87Rb vapor cell operating at 180°C. (C and D) Results of state tomography on initial states after sudden (C) and adiabatic (D) transfers. FFT, fast Fourier transform.

  • Fig. 2 Single-spin independent rotations.

    (A) Schematic diagram of individual spin rotation for 1H (top panel) and 13C (bottom panel), as presented in the text. The initial states of 1H and 13C are aligned to |↑> for simplicity. (B) Combined (top panel) and individual nuclear spin rotation for 13C (middle panel) and 1H (bottom panel). Each data point corresponds to a single measurement. Theoretical fits are shown with solid lines. (C) Clifford-based randomized benchmarking, as described in the main text. (D) Randomized benchmarking results for 13C single-spin control. Each point is an average over 32 random sequences of m Clifford gates, and the error bars indicate the standard error of the mean (note that the vertical axis has a logarithmic scale). A single exponential decay shown with a solid line is used to fit the fidelity decay and reveals an average fidelity of 0.9960(2).

  • Fig. 3 Two-spin CNOT gate.

    (A) Pulse sequences for implementing the CNOT gate. The Uzz operation (see main text) is accomplished with composite pulses. The entire duration of the CNOT gate sequence is 2.7 ms. (B) Output of the CNOT gate applied to the sudden state. (C) Reconstructed CNOT gate in the computational basis. The fidelity of the CNOT gate is 0.9877(2).

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/4/6/eaar6327/DC1

    Supplementary Materials and Methods

    section S1. Hamiltonians and eigenstates

    section S2. State tomography

    section S3. Single-spin control

    section S4. Two-spin control via CNOT gate

    table S1. The evolutions of two-spin operators under the scalar spin-spin coupling.

    table S2. Temporal averaging sequences for state tomography.

    table S3. The evolution of the two-spin operators under the CNOT gate.

    table S4. Reconstructing the CNOT gate based on l-norm.

    fig. S1. Experimental realization of temporal averaging.

    fig. S2. Dependence of NMR signal amplitude on z pulse duration.

  • Supplementary Materials

    This PDF file includes:

    • Supplementary Materials and Methods
    • section S1. Hamiltonians and eigenstates
    • section S2. State tomography
    • section S3. Single-spin control
    • section S4. Two-spin control via CNOT gate
    • table S1. The evolutions of two-spin operators under the scalar spin-spin coupling.
    • table S2. Temporal averaging sequences for state tomography.
    • table S3. The evolution of the two-spin operators under the CNOT gate.
    • table S4. Reconstructing the CNOT gate based on l-norm.
    • fig. S1. Experimental realization of temporal averaging.
    • fig. S2. Dependence of NMR signal amplitude on z pulse duration.

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