Structure, Optimality, and Symmetry in Shadow Unitary Inversion

Reversing unitary operations is a key task in quantum computing and quantum control. In this work, we introduce and develop the framework of shadow unitary inversion, a relaxed variant of unitary inversion in which the goal is to reproduce the action of the inverse unitary only at the level of the expectation value of a fixed observable. This task captures an operational setting in which only shadow information is required and allows query complexities significantly below those of full unitary inversion. We establish a dimension-dependent lower bound showing that any t-query scheme requires t to scale at least linearly with the system dimension, with the constant determined by the spectral properties of the target observable. In the qubit case, we construct a deterministic three-query sequential protocol that achieves exact shadow inversion, and we provide a complete characterization of all admissible qubit channels satisfying the shadow constraint. Numerical evidence suggests that three queries are optimal. For higher-dimensional systems, we develop a semidefinite-programming formulation for optimizing shadow-inversion combs and introduce a representation-theoretic symmetry reduction that decomposes the problem into invariant blocks, substantially reducing the problem size. These results provide the first systematic study for shadow unitary inversion and establish its resource requirements and symmetry structure across dimensions.