Partial Quantum Shadow Tomography for Structured Operators

Abstract

Quantum shadow tomography based on the classical shadow representation provides an efficient way to estimate properties of an unknown quantum state without performing a full quantum state tomography. In scenarios where estimating the expectation values for certain classes of observables is required, obtaining information about the entire density matrix is unnecessary. We propose a partial quantum shadow tomography protocol, which allows the estimation of a subset of the density matrix’s elements contributing to the expectation values of certain classes of observables. This method utilizes tomographically incomplete subsets of single qubit Pauli basis measurements to perform partial tomography, making it experimentally more efficient. We have found a channel description allowing us to extract specific density matrix elements utilizing minimal sets of unitary applications. We demonstrate the advantage over unitary k-designs such as Clifford and full Pauli by numerically analyzing the protocol for random states and different classes of observables. We experimentally demonstrate the partial estimation scheme for a wide class of two-qubit states (pure, entangled, and mixed) in the nuclear magnetic resonance (NMR) platform, which relies on ensemble-based measurements. The full density matrix re-constructed from different partial estimators produces fidelities exceeding 97%