Optimizing quantum tomography via shadow inversion

In quantum information theory, the accurate estimation of observables is pivotal for quantum information processing, playing a crucial role in computational and communication protocols. This work introduces a technique for estimating such objects, leveraging an underutilized resource in the inversion map of classical shadows that greatly refines the estimation cost of target observables without incurring any additional overhead. A generalized framework for computing and optimizing additional degrees of freedom in the homogeneous space of the shadow inversion is given that may be adapted to a variety of near-term problems. In the special case of local measurement strategies, we show feasible optimization leading to an exponential separation in sample complexity versus the standard approach and, in an exceptional case, we give nontrivial examples of optimized postprocessing for local measurements, achieving the same efficiency as the global Cliffords shadows.