Krylov Shadow Tomography: Efficient Estimation of Quantum Fisher Information

Efficiently estimating the quantum Fisher information (QFI) is pivotal in quantum information science but remains an outstanding challenge for large systems due to its high nonlinearity. In this Letter, we tackle this long-standing challenge by integrating the Krylov subspace method-a celebrated tool from applied mathematics-into the framework of shadow tomography. The integrated technique, dubbed Krylov shadow tomography, enables us to formulate a strict hierarchy of nonpolynomial lower bounds on the QFI, among which the highest one matches the QFI exactly. We show that all the bounds can be expressed as expected values of the inverses of Hankel matrices, which are accessible via shadow tomography. Our Krylov shadow tomography therefore opens up a resource-efficient and experimentally feasible avenue to estimate not only nonpolynomial lower bounds but also the QFI itself.