Fermionic partial tomography via classical shadows

We propose a tomographic protocol for estimating any k-body reduced density matrix (k-RDM) of a fermionic state, a ubiquitous step in near-term quantum algorithms for simulating many-body physics, chemistry, and materials. Our approach extends the framework of classical shadows, a randomized approach to learning a collection of quantum state properties, to the fermionic setting. Our sampling protocol employs randomized measurements generated by a discrete group of fermionic Gaussian unitaries, implementable with linear-depth circuits, to achieve near-optimal scaling in the number of repeated state preparations required of fermionic RDM tomography. We also numerically demonstrate that our protocol offers a substantial improvement in constant overheads over prior state-of-the-art for estimating 2-, 3-, and 4-RDMs.