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Complexity and order in approximate quantum error-correcting codes

Nature Physics volume 20, pages 1798–1803 (2024)Cite this article

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Abstract

Some form of quantum error correction is necessary to produce large-scale fault-tolerant quantum computers and finds broad relevance in physics. Most studies customarily assume exact correction. However, codes that may only enable approximate quantum error correction (AQEC) could be useful and intrinsically important in many practical and physical contexts. Here we establish rigorous connections between quantum circuit complexity and AQEC capability. Our analysis covers systems with both all-to-all connectivity and geometric scenarios like lattice systems. To this end, we introduce a type of code parameter that we call subsystem variance, which is closely related to the optimal AQEC precision. For a code encoding k logical qubits in n physical qubits, we find that if the subsystem variance is below an O(k/n) threshold, then any state in the code subspace must obey certain circuit complexity lower bounds, which identify non-trivial phases of codes. This theory of AQEC provides a versatile framework for understanding quantum complexity and order in many-body quantum systems, generating new insights for wide-ranging important physical scenarios such as topological order and critical quantum systems. Our results suggest that O(1/n) represents a common, physically profound scaling threshold of subsystem variance for features associated with non-trivial quantum order.

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Fig. 1: Locality and quantum circuit complexity.
Fig. 2: Schematic circuit complexity phase diagrams of general quantum codes.

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Acknowledgements

We thank A. Anshu, Y.-C. He, H. Ma, S. Sang, B. Yoshida, S. Zhou, Z. Zhou and Y. Zou for valuable discussions and feedback. J.Y. would like to thank A. Burkov for his support. D.G. is partially supported by the National Science Foundation (RQS QLCI grant OMA-2120757). Z.-W.L. is partially supported by a startup funding at YMSC, Tsinghua University. Research at the Perimeter Institute is supported in part 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 Colleges and Universities.

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Authors and Affiliations

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

    Jinmin Yi & Weicheng Ye

  2. Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada

    Jinmin Yi

  3. Joint Center for Quantum Information and Computer Science (QuICS) and Computer Science Department, University of Maryland, College Park, MD, USA

    Daniel Gottesman

  4. Yau Mathematical Sciences Center, Tsinghua University, Beijing, China

    Zi-Wen Liu

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  1. Jinmin Yi
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Contributions

Z.-W.L. conceived and designed this project, drawing inspiration from conversations with J.Y. and W.Y. and especially D.G. J.Y. led the technical development of this work, which was supervised by Z.-W.L. All authors contributed to the discussions that shaped this work. J.Y. and Z.-W.L. are primarily responsible for the technical content and writing of the paper.

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Correspondence to Zi-Wen Liu.

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Yi, J., Ye, W., Gottesman, D. et al. Complexity and order in approximate quantum error-correcting codes. Nat. Phys. 20, 1798–1803 (2024). https://doi.org/10.1038/s41567-024-02621-x

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