Abstract
Reversing unknown quantum dynamics is vital for quantum control and learning, yet full unitary inversion requires a resource-intensive O(d2) queries. Because many applications only require the reversed evolution for a given observable, a critical gap remains in understanding the minimal resources for such targeted reversal. Here, we address this by introducing shadow unitary inversion. We establish a lower bound showing the query complexity must scale at least linearly with system dimension for spectrally biased observables, with the constant determined by the observable’s spectral properties. For qubit case, we construct an explicit, deterministic three-query sequential protocol achieving exact shadow inversion and completely characterize all admissible channels, with numerical evidence suggesting optimality. For higher dimensions, we develop a semidefinite-programming formulation and introduce a representation-theoretic symmetry reduction that decomposes the optimization into invariant blocks, substantially reducing the problem size. Shadow unitary inversion thus offers a resource-efficient path to inverse-dynamics estimation for future quantum control, diagnostics, and learning tasks.
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We thank the two anonymous reviewers for their valuable comments and suggestions, including the example discussed in Supplementary Note 5.
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This work was partially supported by the National Key R&D Program of China (Grant No. 2024YFB4504004), the National Natural Science Foundation of China (Grant Nos. 12447107 and 92576114), the Guangdong Provincial Quantum Science Strategic Initiative (Grant Nos. GDZX2403008 and GDZX2503001), the CCF-Tencent Rhino-Bird Open Research Fund, the Guangdong Provincial Key Lab of Integrated Communication, Sensing and Computation for Ubiquitous Internet of Things (Grant No. 2023B1212010007), the Quantum Science Center of Guangdong-Hong Kong-Macao Greater Bay Area, and the Education Bureau of Guangzhou Municipality.
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Zhen, G., Chen, YA., Jing, M. et al. Structure, optimality, and symmetry in shadow unitary inversion. Commun Phys (2026). https://doi.org/10.1038/s42005-026-02690-9
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DOI: https://doi.org/10.1038/s42005-026-02690-9
