Abstract
A critical challenge in developing scalable quantum systems is correcting the accumulation of errors while performing operations and measurements. It is known that systems where dominant errors can be detected and converted into erasures have relaxed requirements for quantum error correction. Recently, it has been proposed that this can be achieved using a dual-rail encoding of quantum information in the microwave photon states of two superconducting cavities. One necessary step to realize this erasure qubit is to demonstrate a measurement and to flag errors as erasures. In this work, we demonstrate a projective logical measurement of a dual-rail cavity qubit with integrated erasure detection and measure the qubit idling errors. We measure the logical state preparation and measurement errors at the 0.01% level and detect over 99% of the cavity decay events as erasures. We use the precision of this measurement protocol to distinguish different types of error in this system, finding that although decay errors occur with a probability of approximately 0.2% per microsecond, phase errors occur 6 times less frequently and bit flips occur at least 150 times less frequently. These findings represent a confirmation of the expected error hierarchy necessary to concatenate dual-rail cavity qubits into a highly efficient erasure code.
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Data availability
The datasets generated and analysed during this study are available via Zenodo at https://doi.org/10.5281/zenodo.11099521 (ref. 57).
Code availability
The code that supports the findings of this study is available from the corresponding authors upon reasonable request.
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Acknowledgements
We thank the broader mechanical and control system groups at QCI, particularly C. Clothier, R. Chamberlin, M. Maxwell and C. Wehr. This research was supported by the US Army Research Office (ARO) under grant W911NF-23-1-0051, and by the US Department of Energy, Office of Science, National Quantum Information Science Research Centers, Co-design Center for Quantum Advantage (C2QA), under contract no. DE-SC0012704. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing official policies, either expressed or implied, of the ARO or the US government. The US government is authorized to reproduce and distribute reprints for government purpose notwithstanding any copyright notation herein.
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K.S.C., T.S., H.M., T.-C.C., P.L., N.M., A.N. and T.N. designed, implemented and performed the experiments. J.D.T., P.W., S.J.d.G., J.W.O.G., W.D.K., A.M., T.T. and S.H.X. contributed to the design of SPAM protocols and provided key insights into dual-rail coherence error sources. T.S., K.S.C., N.M., J.D.T., J.C.C. and T.T. contributed to the error model design and implementation. J.C.C. and T.T. provided critical insights into the design and optimization of the SPAM protocol. T.K. contributed to the single-qubit gate benchmarking. B.G., C.U.L., G.L., N.K., S.O.M. and J.O.Y. contributed to the design, fabrication and building of the experimental setup as well as initial system validation. A.A., J.C. and L.M.-K. contributed to the implementation of the software system used to perform these experiments. All work was supervised by L.F., J.A., S.P., S.M.G., S.H.M.J. and R.J.S. All authors discussed the results and contributed to the manuscript.
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R.J.S. and L.F. are founders and shareholders of Quantum Circuits, Inc. (QCI). S.P. and S.M.G. receive consulting fees and/or are equity holders in QCI. K.S.C., T.S., H.M., T.-C.C., A.A., J.C., B.G., T.K., N.K., C.U.L., G.L., P.L., L.M.-K., N.M., S.O.M., A.N., T.N., J.O.Y., L.F., J.A., S.H.M.J. and R.J.S. have financial interest in QCI. The US patent ‘Dual-rail qubits based on superconducting resonant cavities and their application for quantum error correction’ has been filed (PCT/US23/84327) by R.J.S., S.M.G., S.P., J.D.T., S.J.d.G., S.H.X., B. Chapman, J.W.O.G., A.M., Y.L., W.D.K., N. Thakur, T.T. and P.W. with Yale University. The remaining authors declare no competing interests.
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Extended data
Extended Data Fig. 1 Decoding measurement outcomes.
(A) Measurement outcomes are decoded and assigned based on the results of each of the check measurements and the logical measurement is the state preparation and measurement protocols. As described in Supplementary Information Section V, there are check measurements associated with the state preparation, and check measurements associated with the logical measurement that come after a delay or an algorithm between the state preparation and measurement. The dashed lines indicate the grouping of the check measurements to state preparation or logical measurement; the colors indicate one of the three categories that are used to assign the outcomes of each of the measurements (B) The outcomes of each of the check or logical measurements can result in the shot being assigned either as FPC (failed state preparation check), FMC (failed measurement check), or FA (failed assignment). Each such k shot is placed into an abstract ‘bucket’, and contributes to the total counts of the outcome, Nk. NFPC counts are removed from the total number of counts, while NFMC and NFA are two of the contributing groups of counts to the erasure fraction.
Extended Data Fig. 2 State assignment data using 2 rounds of measurements.
As in the case of the SPAM experiment with 1 round of measurements, we prepare each of the four dual-rail basis states. Using the results of the 2 rounds of measurements and our decoding strategy, we assign one of the four states {‘00’, ‘01’, ‘10’, ‘11’}, or the ambiguous outcome ‘A’ in the case where the two measurement outcomes disagree. Results of preparing the two logical state \(\left\vert 01\right\rangle\) and \(\left\vert 10\right\rangle\) are shown in the left two panels; results of intentionally preparing the leakage states \(\left\vert 00\right\rangle\) and \(\left\vert 11\right\rangle\) are shown in the right two panels. The green horizontal lines simulation results modeling the SPAM experiment with two-rounds of measurements. For each state, we repeat the experiment a total of 100, 000 times. The height of the each bar and center of the black line correspond to the mean of the data and error bars represent the standard error, showing ± 1σ.
Extended Data Fig. 3 Short-time Ramsey experiment analysis.
(A) For these experiments, we perform a modified version of the Ramsey experiment described in the Main Text. Here we sweep both the Ramsey delay as well as the phase of the second π/2 pulse. (B) Top left: Logical results showing Ramsey oscillations for each delay out to 20 μs. The markers correspond to experimental results while the lines correspond to the fit. Bottom left: Fit residuals. Right: Extracted Ramsey contrast (valued between 0 and 1) as a function of Ramsey delay. Each marker corresponds to a different delay, color-coded to the data in the left column. Straight black line is a linear fit to the data.
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Chou, K.S., Shemma, T., McCarrick, H. et al. A superconducting dual-rail cavity qubit with erasure-detected logical measurements. Nat. Phys. 20, 1454–1460 (2024). https://doi.org/10.1038/s41567-024-02539-4
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DOI: https://doi.org/10.1038/s41567-024-02539-4
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