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. 2020 Jan 30;11(1):587.
doi: 10.1038/s41467-020-14376-z.

Error-mitigated Quantum Gates Exceeding Physical Fidelities in a Trapped-Ion System

Free PMC article

Error-mitigated Quantum Gates Exceeding Physical Fidelities in a Trapped-Ion System

Shuaining Zhang et al. Nat Commun. .
Free PMC article


Various quantum applications can be reduced to estimating expectation values, which are inevitably deviated by operational and environmental errors. Although errors can be tackled by quantum error correction, the overheads are far from being affordable for near-term technologies. To alleviate the detrimental effects of errors on the estimation of expectation values, quantum error mitigation techniques have been proposed, which require no additional qubit resources. Here we benchmark the performance of a quantum error mitigation technique based on probabilistic error cancellation in a trapped-ion system. Our results clearly show that effective gate fidelities exceed physical fidelities, i.e., we surpass the break-even point of eliminating gate errors, by programming quantum circuits. The error rates are effectively reduced from (1.10 ± 0.12) × 10-3 to (1.44 ± 5.28) × 10-5 and from (0.99 ± 0.06) × 10-2 to (0.96 ± 0.10) × 10-3 for single- and two-qubit gates, respectively. Our demonstration opens up the possibility of implementing high-fidelity computations on a near-term noisy quantum device.

Conflict of interest statement

The authors declare no competing interests.


Fig. 1
Fig. 1. Paradigm of error-mitigated quantum computation.
a Quantum black box based on a trapped 171Yb+-ion system. Each button on the surface corresponds to an operation exerted on the quantum system encapsulated, where the buttons with ρ and M are for initial state preparation and computational basis measurement, whose results are indicated by the lights. The other buttons are for single-qubit and two-qubit quantum operations on certain qubits. The operations are implemented by global (blue) and individual (purple) laser beams illuminating the ion qubits. b Characterization of the quantum black box. The error-affected state preparation and measurement is characterized by the Gram matrix g and the effect of each operation G, such as Yπ2 and MSYY, is described by a Pauli transfer matrix RG in the superoperator formalism, which is obtained by gate set tomography. c Construction of unbiased estimator of an expectation value specified by a quantum circuit, with building blocks including initial state preparation, different single-qubit and two-qubit gates, and the final measurement. With error mitigation, the distribution of the output expectation value is shifted towards the ideal value at a cost of enlarged variance. d Quasi-probability decomposition for the ideal initial state and exemplary single-qubit and two-qubit gates. As the errors in state preparation and those in measurement are indistinguishable, we ascribe both of the errors to state preparation and decompose the ideal initial state with a set of experimental basis states, prepared by state initialization followed by a random fiducial gate. The PTM of an ideal quantum gate can be expanded as a quasi-probability distribution over random gate sequences consisting of the experimental gate and one of the experimental basis operations, Pauli operations in the experiment. The error-mitigated estimation of the expectation value is then obtained by the Monte-Carlo method (see Methods).
Fig. 2
Fig. 2. Characterization of noisy quantum devices obtained by gate set tomography.
a Gram matrix and b PTMs of single-qubit gates for the single-qubit case. The single-qubit experiments are implemented with a single trapped ion. The Gram matrix characterize the SPAM error, which is obtained by preparing states in S1 and measuring expectation values of operators in P1. Here we show the PTMs of experimental gates X±π2 and Y±π2, the so-called computational gates in randomized benchmarking, as examples (PTMs of other experimental gates are shown in Supplementary Fig. 1). c Pauli transfer matrices of the experimental gates MSYY and MSZZ in the two-qubit case. It is worth noting that we calibrate the SPAM errors as proposed in ref. and the PTMs of single-qubit gates on both qubits (not shown) are not noticeably different to those for the single-qubit case. In each subfigure, the left column shows the experimentally obtained matrices and the right column shows the difference between the experimental and the ideal matrices, i.e., RGRGid with G being one of the quantum operations being characterized.
Fig. 3
Fig. 3. Quasi-probability decomposition.
a Quasi-probabilities in the decomposition of the ideal single-qubit initial state with experimental initial states in S1. b Quasi-probabilities in the decomposition of the inverse noise operations of the four experimental single-qubit gates {X±π2,Y±π2}. c The same as b for the experimental two-qubit gates {MSYY, MSZZ}.
Fig. 4
Fig. 4. Experimental data for error-mitigated quantum computation.
a The single-qubit randomized benchmarking. The data points with (purple square) and without (yellow diamond) error mitigation are obtained from averaging the final-state fidelitiy over different random sequences of the same length (black dots). The error bars are the SD of the average fidelities computed using the formula of uncertainty propagation. The solid lines, obtained by fitting, show the exponential decrease of the average fidelities, indicating the physical and effective average errors per gate being (1.10 ± 0.12) × 10−3 and (1.44 ± 5.28) × 10−5, respectively. Please note that some of the fidelities with error mitigation are larger than 1 because of the rescaling factor C > 1 (see main text and Methods) and the limited sampling for data points. Although the current protocol does not guarantee a physical outcome, the error mitigation procedure shifts the distribution of the computation result towards the true value with a large enough sampling. b The two-qubit random-circuit computation. Decay rates indicated by the average fidelity curves without and with error mitigation are (0.99 ± 0.06) × 10−2 and (0.96 ±  0.10) × 10−3, respectively.

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