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. Reversing unitary operations is crucial in quantum computing and control, yet remains a challenging task. The authors introduce shadow unitary inversion, achieving efficient inversion at the expectation value level with reduced query complexity, offering a promising approach for quantum error correction and information recovery.

Nature is a scientific journal publishing research articles, reviews, and commentary across a wide range of scientific disciplines. Readers can find articles on topics such as biology, chemistry, physics, and environmental science. Additionally, they can learn about  research, scientific discoveries, and advances in understanding the natural world.

Nature

Shadow unitary inversion is introduced as a resource-efficient method for reversing unknown quantum dynamics at the expectation value level, without requiring full unitary inversion. A lower bound on query complexity is established, showing it must scale at least linearly with system dimension for spectrally biased observables. For the qubit case, an explicit deterministic three-query sequential protocol achieves exact shadow inversion, with all admissible channels fully characterized. For higher dimensions, a semidefinite-programming formulation with representation-theoretic symmetry reduction substantially reduces computational complexity. The approach offers a promising path for quantum error correction, information recovery, and quantum learning tasks.

Structure, optimality, and symmetry in shadow unitary inversion