State preservation by repetitive error detection in a superconducting quantum circuit

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Quantum computing becomes viable when a quantum state can be protected from environment-induced error. If quantum bits (qubits) are sufficiently reliable, errors are sparse and quantum error correction (QEC)1, 2, 3, 4, 5, 6 is capable of identifying and correcting them. Adding more qubits improves the preservation of states by guaranteeing that increasingly larger clusters of errors will not cause logical failure—a key requirement for large-scale systems. Using QEC to extend the qubit lifetime remains one of the outstanding experimental challenges in quantum computing. Here we report the protection of classical states from environmental bit-flip errors and demonstrate the suppression of these errors with increasing system size. We use a linear array of nine qubits, which is a natural step towards the two-dimensional surface code QEC scheme7, and track errors as they occur by repeatedly performing projective quantum non-demolition parity measurements. Relative to a single physical qubit, we reduce the failure rate in retrieving an input state by a factor of 2.7 when using five of our nine qubits and by a factor of 8.5 when using all nine qubits after eight cycles. Additionally, we tomographically verify preservation of the non-classical Greenberger–Horne–Zeilinger state. The successful suppression of environment-induced errors will motivate further research into the many challenges associated with building a large-scale superconducting quantum computer.

At a glance


  1. Repetition code: device and algorithm.
    Figure 1: Repetition code: device and algorithm.

    a, The repetition code is a one-dimensional (1D) variant of the surface code, and is able to protect against (bit-flip) errors. The code is implemented using an alternating pattern of data and measurement qubits. b, Optical micrograph of the superconducting quantum device, consisting of nine Xmon21 transmon qubits with individual control and measurement, with a nearest-neighbour coupling scheme. c, The repetition code algorithm uses repeated entangling and measurement operations which detect bit-flips, using the parity scheme on the right. Using the output from the measurement qubits during the repetition code for error detection, the initial state can be recovered by removing physical errors in software. Measurement qubits are initialized into the |0right fence state and need no reinitialization as measurement is QND.

  2. Error propagation and identification.
    Figure 2: Error propagation and identification.

    a, The quantum circuit for three cycles of the repetition code, and examples of errors. Errors propagate horizontally in time, and vertically through entangling gates. Different errors lead to different detection patterns: an error on a measurement qubit (gold) is detected in two subsequent rounds. Data qubit errors (purple, red, blue) are detected on neighbouring measurement qubits in the same or next cycle. Data errors after the last round (blue) are detected by constructing the final set of eigenvalues from the data qubit measurements. b, The connectivity graph for the quantum circuit above, showing measurements and possible patterns of detection events (grey), see main text for details. The example detection events and their connections are highlighted, and the corresponding detected errors are shown on the right, which when applied, will recover the input data qubit state.

  3. Protecting the GHZ state from bit-flip errors.
    Figure 3: Protecting the GHZ state from bit-flip errors.

    a, Quantum circuit for generating the GHZ state and two cycles of the repetition code. CNOT gates are physically implemented with controlled-phase (CZ) and single qubit gates. b, Quantum state tomography on the input (top left ‘Input’, left of black dashed line), and after the repetition code conditional on the detection events (between black dashed lines): we input a GHZ state with a fidelity (F) of 82%, and find, for the case of no detection events (top right ‘Output’, above grey dashed line), a 78% fidelity GHZ state. For the detection event connecting both measurement qubits (bottom left ‘Raw output’, below grey dashed line), indicating a likely bit-flip error on the central data qubit, we find that through correcting in post-processing by exchanging matrix elements we recover the major elements of the diagonal (bottom right ‘Corrected output’). We also recover non-zero off-diagonal elements, indicating some bit-flip errors are coherent. Real parts are shown, |Im(ρ)| < 0.03.

  4. Logical state preservation with the repetition code.
    Figure 4: Logical state preservation with the repetition code.

    a, Information flowchart of the repetition code. The data qubits are initialized into |0Lright fence or |1Lright fence, and the repetition code is repeated k times. In post-processing, the measurement qubit outcomes are converted into detection events and matched to find likely errors, see Fig. 2. A successful recovery converts the measured data qubit state into the input state. b, Memory fidelity versus time and cycles for a single physical qubit (black) and the five- (blue) and nine- (red) qubit repetition code. Note that energy relaxation decays from a fidelity of 1 to 0, whereas the repetition code decays from a fidelity of 1 to 0.5. Five qubit data sampled from nine qubit data, see Supplementary Information. The average physical qubit lifetime (‘data qubit avg.’) is T1 = 29 µs, and after eight cycles we see an improvement in error rate by a factor of 2.7 (blue arrow at right) for five qubits (‘5 qubit RC’), and 8.5 (red arrow at right) for nine qubits (‘9 qubit RC’) when using the repetition code. This indicates a Λ parameter of 3.2 (green arrow) for our system after eight cycles. c, Average number of detection events per measurement qubit (open symbols), versus cycle number, for experiments consisting of eight cycles. We see an increasing average rate of detection events (black line) with increasing cycle number. This can be attributed to the statistically increasing number of odd parity measurements, see text. Grey regions indicate initialization and final data qubit measurement.


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Author information

  1. These authors contributed equally to this work.

    • J. Kelly,
    • R. Barends &
    • A. G. Fowler
  2. Present address: Google Inc., Santa Barbara, California93117, USA.

    • R. Barends,
    • A. G. Fowler,
    • E. Jeffrey,
    • D. Sank,
    • J. Y. Mutus,
    • Yu Chen,
    • P. Roushan &
    • John M. Martinis


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

    • J. Kelly,
    • R. Barends,
    • A. G. Fowler,
    • A. Megrant,
    • E. Jeffrey,
    • T. C. White,
    • D. Sank,
    • J. Y. Mutus,
    • B. Campbell,
    • Yu Chen,
    • Z. Chen,
    • B. Chiaro,
    • A. Dunsworth,
    • I.-C. Hoi,
    • C. Neill,
    • P. J. J. O’Malley,
    • C. Quintana,
    • P. Roushan,
    • A. Vainsencher,
    • J. Wenner,
    • A. N. Cleland &
    • John M. Martinis
  2. Centre for Quantum Computation and Communication Technology, School of Physics, The University of Melbourne, Victoria 3010, Australia

    • A. G. Fowler
  3. Department of Materials, University of California, Santa Barbara, California 93106, USA

    • A. Megrant


J.K. and R.B. designed the sample and performed the experiment. A.G.F. and J.M.M. designed the experiment. J.K., R.B. and A.M. fabricated the sample. A.G.F., J.K. and R.B. analysed the data. J.K., R.B., A.G.F. and J.M.M. co-wrote the manuscript. All authors contributed to the fabrication process, experimental set-up and manuscript preparation.

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The authors declare no competing financial interests.

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Supplementary information

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  1. Supplementary Information (2.8 MB)

    This file contains Supplementary Text and Data, Supplementary Figures 1-31, Supplementary Tables 1-3 and additional references.

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