Research ArticlePHYSICS

Quantum computation with universal error mitigation on a superconducting quantum processor

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Science Advances  06 Sep 2019:
Vol. 5, no. 9, eaaw5686
DOI: 10.1126/sciadv.aaw5686
  • Fig. 1 Illustration of the quantum error mitigation protocol implemented on a superconducting quantum device.

    (A) Flowchart of the universal quantum error mitigation, which has two stages: GST and the random circuit computation. (B) Layout of four qubits actively used in the experiment. The information is encoded in Q1 and Q2, and the other two qubits QA1 and QA2 are ancillary qubits. (C) Controlled-ϕ-phase gate Cϕ realized using the dressed state gate Uϕ and single-qubit gates. The single-qubit gate Pθ=eiθ2P, where P = X, Y, Z. (D) Reset gate R∣ψ〉 realized using an ancillary qubit QA1 or QA2, which reinitializes the qubit Q1 or Q2 in the state ∣ψ〉 = G∣0〉.

  • Fig. 2 GST circuits and data.

    (A) Circuits for one- and two-qubit GST. The gate to be characterized (marked in gray) is implemented in between the state preparation and measurement. Gram matrices and matrices of measurement-initialization gates are obtained using the one-qubit circuit. Matrices of two-qubit gates are obtained using the two-qubit circuit. For the gram matrix, the gate is null. (B) Gram matrix gexp of the qubit Q1. (C) PTMs of the two-qubit gate Cπ. For the ideal gate, each element is calculated as Uij=Tr[σiUσj]/4, where U is the ideal superoperator for Cπ. For the experiment gate, the matrix Û is the result of GST.

  • Fig. 3 Schematics and results for one- and two-qubit PEC experiments.

    (A) Circuit of the one-qubit computation. In PEC, the measurement of the observable Z is replaced by random gates. (B) Random circuit of the one-qubit computation, in which the measurement in the original circuit is replaced by the measurement of the observable σi. (C) Circuit of the two-qubit DQCp computation. In PEC, the two-qubit gate and the measurement are replaced by random operations. (D) Circuit of the Pauli twirling. (E) Representative random circuits of the two-qubit computation. μ denotes the outcome of the corresponding measurement, and w is the weight of the corresponding circuit wj,i as defined in the main text (in the figure, the subscript of w denotes the number of the instance). N is the total number of instances. The circuit in the blue box is the replacement of the two-qubit gate Cϕ. We note that, in instance 1, four single-qubit gates are Pauli gates of the Pauli twirling. (F) Results of the one-qubit computation. The probability distribution of the computation result is plotted. Without error, the ideal result is 〈Z〉 = 0. (G) Results of the two-qubit computation. Each data point is obtained using 1,000,000 instances. We implement random circuits 10,000 times to compute one average value of X and repeat the computation to obtain 100 average values. The error bar indicates the SD of these average values.

  • Table 1 Sixteen single-qubit basis operations.

    Pθ=eiθ2P denotes the gate of rotation along the P axis by an angle of θ, where P = X, Y, Z. MP denotes the operation of measuring the eigenvalue of the Pauli operator P whose outcomes are ±1. MI+P2 and MP are the same operation, but outcomes are noted differently, and MI+P2 denotes the operation of measuring the eigenvalue of the operator I+P2 whose outcomes are 0 and 1. R∣ψ〉 denotes the operation of resetting the qubit state to ∣ψ〉. For composed operations, operations are implemented from left to right in sequence. These basis operations are linearly independent and complete; therefore, all single-qubit operations can be decomposed as linear combinations of basis operations. Non-unital operations, i.e., reset gates, are necessary in the basis set to efficiently decompose the non-unital part of an operation. The basis set minimizing the variance of the computation result is preferred.

    No.OperationNo.Operation
    1I9Xπ,Yπ2
    2Xπ10Yπ,Xπ2
    3Yπ11MI+X2,R0+1
    4Zπ12MI+X2,R01
    5Xπ213MI+Y2,R0+i1
    6Yπ214MI+Y2,R0i1
    7Zπ215MI+Z2,R0
    8Xπ,Zπ216MI+Z2,R1

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/9/eaaw5686/DC1

    Randomized benchmarking for single-qubit gates

    Heating rate measurement

    Readout error for Q1 and Q2

    One-qubit QEM experiment

    Measurement and reset gates

    Decomposing Cϕ gate

    Depolarizing error channels

    Fig. S1. Randomized benchmarking data.

    Fig. S2. Heating rate measurement.

    Fig. S3. One-qubit QEM experiment.

    Fig. S4. Measurement-reset gate.

    Fig. S5. Decomposition of the Cπ gate.

    Table S1. Error rates of readout measured by repeatedly preparing the state ∣0〉 or ∣1〉 and measuring the probability of incorrect output.

    Table S2. Gate fidelities for all measurement-reset gates used in this paper.

  • Supplementary Materials

    This PDF file includes:

    • Randomized benchmarking for single-qubit gates
    • Heating rate measurement
    • Readout error for Q1 and Q2
    • One-qubit QEM experiment
    • Measurement and reset gates
    • Decomposing Cϕ gate
    • Depolarizing error channels
    • Fig. S1. Randomized benchmarking data.
    • Fig. S2. Heating rate measurement.
    • Fig. S3. One-qubit QEM experiment.
    • Fig. S4. Measurement-reset gate.
    • Fig. S5. Decomposition of the Cπ gate.
    • Table S1. Error rates of readout measured by repeatedly preparing the state ∣0〉 or ∣1〉 and measuring the probability of incorrect output.
    • Table S2. Gate fidelities for all measurement-reset gates used in this paper.

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