Structure, Optimality, and Symmetry in Shadow Unitary Inversion

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Abstract
The ability to reverse any unknown unitary operation plays a fundamental role in quantum computing. While existing studies mostly focus on realizing the inversion map of the unknown unitary, how to reverse a unitary with respect to a given observable, which we call shadow unitary inversion, has remained a natural basic question that is less developed. In this work, we systematically investigate shadow unitary inversion by providing explicit protocols and optimization problem simplification. First, we present a deterministic protocol for shadow inversion of qubit-unitaries. Such construction sequentially queries the unitary 3 times, which is suggested to be optimal by our numerical experiments. Second, we provide a complete characterization of feasible quantum operations for qubit shadow inversion under any fixed qubit observable. Third, for the qudit case, we give a framework of semidefinite programming for optimizing the shadow unitary inversion sequential protocol for tackling high-dimensional cases, utilizing tools from representation theory.
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