# Error-transparent operations on a logical qubit protected by quantum error correction

## Abstract

Universal quantum computation1 is striking for its unprecedented capability in processing information, but its scalability is challenging in practice because of the inevitable environment noise. Although quantum error correction (QEC) techniques2,3,4,5,6,7,8 have been developed to protect stored quantum information from leading orders of error, the noise-resilient processing of the QEC-protected quantum information is highly demanded but remains elusive9. Here, we demonstrate phase gate operations on a logical qubit encoded in a bosonic oscillator in an error-transparent (ET) manner. Inspired by refs. 10,11, the ET gates are extended to the bosonic code and are able to tolerate errors on the logical qubit during gate operations, regardless of the random occurrence time of the error. With precisely designed gate Hamiltonians through photon-number-resolved a.c. Stark shifts, the ET condition is fulfilled experimentally. We verify that the ET gates outperform the non-ET gates with a substantial improvement of gate fidelity after an occurrence of the single-photon-loss error. Our ET gates in superconducting quantum circuits can be readily extended to multiple encoded qubits and a universal gate set is within reach, holding the potential for reliable quantum information processing.

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## Data availability

Source data for Figs. 24 are available with the paper. All other data relevant to this study are available from the corresponding author upon reasonable request.

## Code availability

The code used for simulations is available from the corresponding authors upon reasonable request.

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## Acknowledgements

This work was supported by the National Key Research and Development Program of China (grant 2017YFA0304303) and the National Natural Science Foundation of China (grants 11925404 and 11874235). C.-L.Z. was supported by the National Natural Science Foundation of China (grants 11874342 and 11922411) and the Anhui Initiative in Quantum Information Technologies (AHY130200).

## Author information

Authors

### Contributions

Y.M. performed the experiment and analysed the data with the assistance of Y.X. L.S. directed the experiment. Y.M. and C.-L.Z. proposed the experiment. C.-L.Z. provided theoretical support. L.H. developed the field-programmable gate array logic. W.C. fabricated the Josephson parametric amplifier. X.Y., X.M. and W.W. fabricated the devices with the assistance of X.P., H.W. and Y.P.S. Y.M., C.-L.Z. and L.S. wrote the manuscript with feedback from all authors.

### Corresponding authors

Correspondence to C.-L. Zou or L. Sun.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature Physics thanks Tanay Roy and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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## Supplementary information

### Supplementary information

Supplementary Figs. 1–8, Tables 1–4 and Discussion.

## Source data

### Source Data Fig. 2

Data of Fig. 2c, including standard deviation as the error bars.

### Source Data Fig. 3

Data of Fig. 3b–d, including standard deviation as the error bars.

### Source Data Fig. 4

Data of Fig. 4b, including standard deviation as the error bars.

## Rights and permissions

Reprints and Permissions

Ma, Y., Xu, Y., Mu, X. et al. Error-transparent operations on a logical qubit protected by quantum error correction. Nat. Phys. 16, 827–831 (2020). https://doi.org/10.1038/s41567-020-0893-x

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