Research ArticleAPPLIED SCIENCES AND ENIGINEERING

Demonstration of universal parametric entangling gates on a multi-qubit lattice

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Science Advances  02 Feb 2018:
Vol. 4, no. 2, eaao3603
DOI: 10.1126/sciadv.aao3603
  • Fig. 1 Device architecture.

    (A) Optical image of the eight-qubit superconducting circuit, consisting of four fixed-frequency (Q0, Q2, Q4, Q6) and four flux-tunable transmon qubits (Q1, Q3, Q5, Q7), used in the experiments. The inset shows a zoomed-in version of one of the tunable qubits. The dimensions of the chip are 5.5 mm × 5.5 mm. (B) Circuit schematics of a chain of three qubits on the chip, where QF represents the fixed transmons and QT represents the tunable transmons. Each tunable qubit has a dedicated flux bias line connected to ac and dc drives combined using a bias tee, which tunes the time-dependent magnetic flux Φ(t) threaded through its asymmetric SQUID loop, as depicted by the arrows.

  • Fig. 2 Parametrically activated entangling interactions.

    (A) Under modulation, coherent population exchange is observed within the |0〉 ↔ |1〉 subspace of Q0 (left) and within the |1〉 ↔ |2〉 subspace of Q1 (right). Excited-state visibility axes are the averaged heterodyne signal of the readout pulse along an optimal IQ quadrature axis, scaled to the separation in the IQ space of the attractors associated with ground and excited states of the qubits. Inset: Energy level diagrams of the |11〉 ↔ |02〉 transition of Q0 and Q1. (B) Data from the dashed line in (A) show the time-domain evolution between Q0 and Q1 on resonance, as teal (circles) and pink (triangles), respectively, allowing the identification of the target modulation duration of one period (τ = 278ns). (C) Determination of entangling-phase accumulation for the tunable qubit Q1. Inset: Circuit diagram of the Ramsey interferometer used to detect a geometric phase.

  • Fig. 3 Quantum process tomography.

    Process matrices of (A) the ideal process and CZ gates between (B) Q0-Q1, (C) Q1-Q2, and (D) Q2-Q3. The achieved average fidelities are measured to be 95, 93, and 91%, respectively.

  • Fig. 4 QST of GHZ state.

    (A) Quantum algorithm used to prepare the state Embedded Image using CZ gates and the QST routine used to estimate the resulting density matrix. (B) Reconstructed density matrix of the prepared GHZ estimated from QST. The resulting state fidelity is estimated to be ℱ = 79%, in agreement with the expected performance of the three individual CZ gates, with color encoding the complex phase of each element. Density matrix elements below |ρnm| ≤ 0.01 are cast transparent for visibility.

  • Fig. 5 Cross-talk.

    (A) Pulse sequences used for quantifying the effect of cross-talk from ancilla qubits on the performance of CZ gates. To do this, first, an arbitrary bitstring register of six ancilla qubits is prepared, with each qubit in either the ground or excited state. Then, process tomography is performed on the CZ gate between the other two qubits on the eight-qubit chip to extract a fidelity. (B) Histogram of the estimated infidelities measured using this algorithm. (C) Average process fidelities achieved as a function of the number of excited qubits in the ancilla register.

  • Table 1 Characteristic parameters of the eight-qubit device.

    ωr represents the frequency of the resonator, Embedded Image is the qubit frequency (at zero flux), Embedded Image is the frequency of the flux-tunable qubit at Embedded Image, η is the anharmonicity of the qubit, T1 is the energy relaxation time of the qubit, Embedded Image is the Ramsey phase coherence time, ℱRO is the single-shot readout assignment fidelity, and p is the single-qubit gate average error probability estimated as the decay of polarization under randomized benchmarking with Pauli generators of the Clifford group. Note that the anharmonicities of the flux-tunable qubits are measured at their operating frequencies.

    Qubit
    index
    ωr/2π
    (MHz)
    Embedded Image
    (MHz)
    Embedded Image
    (MHz)
    − η/2π
    (MHz)
    T1
    (μs)
    Embedded Image
    (μs)
    RO
    (%)
    p
    (%)
    Q05065.03719.1216.234.118.195.01.43
    Q15278.04934.03817.9204.017.04.393.20.70
    Q25755.04685.8199.414.212.993.71.02
    Q35546.04870.93830.0204.015.86.690.00.37
    Q45164.04031.5211.023.718.795.2*0.70
    Q55457.34817.63920.0175.228.011.787.3*2.00
    Q65656.84662.5196.616.915.493.8*1.20
    Q75388.14812.43803.5182.85.68.689.9*1.35

    *Non-quantum nondemolition readout (49).

    • Table 2 Characteristics of the two-qubit CZ gates performed between neighboring qubit pairs (Q0,Q1), (Q1,Q2), and (Q2,Q3).

      gn represents the effective qubit-qubit coupling under modulation, ωm is the qubit modulation frequency, δω is the tunable qubit frequency shift under modulation, τ is the duration of the CZ gate, and ℱQPT is the two-qubit gate fidelity measured by QPT. The theoretical tunable qubit frequency shifts under modulation (δωth/2π) were obtained analytically using the experimentally determined modulation frequencies ωm and are very close to the experimentally measured values (δω/2π). The gate durations and effective qubit-qubit couplings include pulse risetimes of 40 ns to suppress the effect of pulse turn-on phase.

      Qubitsgn/2π
      (MHz)
      ωm/2π
      (MHz)
      δωth/2π
      (MHz)
      δω/2π
      (MHz)
      τ
      (ns)
      QPT
      (%)
      Q0 - Q12.538327028127895
      Q1 - Q21.838632333035393
      Q2 - Q31.5920025725739591
    • Table 3 Error analysis for the two-qubit CZ gate between pairs (Q1,Q2).

      Contributions to the average infidelity estimated from QPT for several error channels.

      Error channel or processContribution to average
      infidelity bound (≤%)
      Decoherence6.5
      State preparation and measurement (SPAM) error0.2
      Tomography rotations2.0
      Leakage into |02〉6.0
      Residual ZZ coupling1.9
      Spurious sidebands0.03
      Instrumentation drift1.0

    Supplementary Materials

    • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/4/2/eaao3603/DC1

      Fabrication and design

      Theoretical predictions of the gate parameters

      Single-shot readout

      Quantum process tomography

      State tomography

      Process tomography

      Cross-talk QPT

      fig. S1. Steps of the fabrication process of the eight-qubit quantum processor.

      fig. S2. Theoretical predictions for activating parametric gates between a linear chain of three qubits.

      fig. S3. Raw data of readout classifier.

      fig. S4. Performance of the individual trained readout classifier for a qubit.

      table S1. Characteristics of the two-qubit CZ gates performed between neighboring qubit pairs (Q0,Q1), (Q1,Q2), and (Q2,Q3).

      table S2. Averaged quantum process fidelity for different preparation states for the register of six ancilla qubits.

      References (5055)

    • Supplementary Materials

      This PDF file includes:

      • Fabrication and design
      • Theoretical predictions of the gate parameters
      • Single-shot readout
      • Quantum process tomography
      • State tomography
      • Process tomography
      • Cross-talk QPT
      • fig. S1. Steps of the fabrication process of the eight-qubit quantum processor.
      • fig. S2. Theoretical predictions for activating parametric gates between a linear chain of three qubits.
      • fig. S3. Raw data of readout classifier.
      • fig. S4. Performance of the individual trained readout classifier for a qubit.
      • table S1. Characteristics of the two-qubit CZ gates performed between neighboring qubit pairs (Q0,Q1), (Q1,Q2), and (Q2,Q3).
      • table S2. Averaged quantum process fidelity for different preparation states for the register of six ancilla qubits.
      • References (50–55)

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