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Experimental super-Heisenberg quantum metrology with indefinite gate order

Nature Physics volume 19, pages 1122–1127 (2023)Cite this article

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Abstract

The precision of quantum metrology is widely believed to be restricted by the Heisenberg limit, corresponding to a root mean square error that is inversely proportional to the number of independent processes probed in an experiment, N. In the past, some proposals have challenged this belief, for example, using nonlinear interactions among the probes. However, these proposals turned out to still obey the Heisenberg limit with respect to other relevant resources, such as the total energy of the probes. Here we present a photonic implementation of a quantum metrology protocol surpassing the Heisenberg limit by probing two groups of independent processes in a superposition of two alternative causal orders. Each process creates a phase-space displacement, and our setup is able to estimate a geometric phase associated with two sets of N displacements with an error that falls quadratically with N. Our results only require a single-photon probe with an initial energy that is independent of N. Using a superposition of causal orders outperforms every setup where the displacements are probed in a definite order. Our experiment features the demonstration of indefinite causal order in a continuous-variable system, and opens up the experimental investigation of quantum metrology setups boosted by indefinite causal order.

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Fig. 1: Geometric phase associated with two sets of phase-space displacements.
Fig. 2: Experimental setup for coherently controlling gate orders in a continuous-variable photonic system.
Fig. 3: P− for varying N.
Fig. 4: Super-Heisenberg scaling of the RMSE of A.
Fig. 5: Scaling of the total phase against N.

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Data availability

The data that support the findings of this study are presented in the article and Supplementary Information, and are available from the corresponding authors upon reasonable request. Source data are provided with this paper.

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Acknowledgements

This work was supported by the Innovation Program for Quantum Science and Technology (no. 2021ZD0301200), National Natural Science Foundation of China (grant nos. 12122410, 92065107, 11874344, 61835004, 11821404), Anhui Initiative in Quantum Information Technologies (AHY060300), the Fundamental Research Funds for the Central Universities (grant nos. WK2030000038, WK2470000034), the Hong Kong Research Grant Council through grant 17300918 and through the Senior Research Fellowship Scheme SRFS2021-7S02, the Croucher Foundation, and the ID No. 62312 Grant from the John Templeton Foundation, as part of The Quantum Information Structure of Spacetime Project (QISS, https://www.qiss.fr/). Research at the Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development, Canada, and by the Province of Ontario through the Ministry of Research, Innovation and Science. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation.

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Author notes

  1. Yong-Jian Han, Bi-Heng Liu, Jin-Shi Xu, Geng Chen, Chuan-Feng Li & Guang-Can Guo

    Present address: Hefei National Laboratory, University of Science and Technology of China, Hefei, China

  2. These authors contributed equally: Peng Yin, Xiaobin Zhao.

Authors and Affiliations

  1. CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, China

    Peng Yin, Yu Guo, Wen-Hao Zhang, Gong-Chu Li, Yong-Jian Han, Bi-Heng Liu, Jin-Shi Xu, Geng Chen, Chuan-Feng Li & Guang-Can Guo

  2. CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, China

    Peng Yin, Yu Guo, Wen-Hao Zhang, Gong-Chu Li, Yong-Jian Han, Bi-Heng Liu, Jin-Shi Xu, Geng Chen, Chuan-Feng Li & Guang-Can Guo

  3. QICI Quantum Information and Computation Initiative, Department of Computer Science, The University of Hong Kong, Hong Kong, China

    Xiaobin Zhao, Yuxiang Yang & Giulio Chiribella

  4. Institute for Theoretical Physics, ETH Zürich, Zürich, Switzerland

    Yuxiang Yang

  5. Department of Computer Science, University of Oxford, Oxford, UK

    Giulio Chiribella

  6. Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada

    Giulio Chiribella

  7. Hefei National Laboratory, University of Science and Technology of China, Hefei, China

    Gong-Chu Li

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Contributions

P.Y., X.Z., Y.Y. and G. Chiribella proposed the framework of the theory and made the calculations. P.Y., W.-H.Z., G.-C.L. and G. Chen planned and designed the experiment. P.Y. and G. Chen carried out the experiment assisted by B.-H.L., J.-S.X., Y.-J.H. and Y.G. P.Y., G. Chen, X.Z., Y.Y. and G. Chiribella analysed the experimental results and wrote the manuscript. G.-C.G., C.-F.L. and G. Chiribella supervised the project. All the authors discussed the experimental procedures and results.

Corresponding authors

Correspondence to Giulio Chiribella, Geng Chen or Chuan-Feng Li.

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Supplementary Notes 1 and 2 and Fig. 1.

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Yin, P., Zhao, X., Yang, Y. et al. Experimental super-Heisenberg quantum metrology with indefinite gate order. Nat. Phys. 19, 1122–1127 (2023). https://doi.org/10.1038/s41567-023-02046-y

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