Fermionic Partial Tomography via Classical Shadows

TL;DR: This work proves that estimating all k-RDM elements to additive precision ϵ requires on the order of (n/k)k^{3/2}log(n)/ϵ^{2} repeated state preparations, which is optimal up to the logarithmic factor.

Abstract: We propose a tomographic protocol for estimating any $k$-body reduced density matrix ($k$-RDM) of an $n$-mode 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 uses randomized measurement settings generated by a discrete group of fermionic Gaussian unitaries, implementable with linear-depth circuits. We prove that estimating all $k$-RDM elements to additive precision $ϵ$ requires on the order of $(\genfrac{}{}{0ex}{}{n}{k}){k}^{3/2}\mathrm{log}(n)/{ϵ}^{2}$ repeated state preparations, which is optimal up to the logarithmic factor. Furthermore, numerical calculations show that our protocol offers a substantial improvement in constant overheads for $k\ensuremath{\ge}2$, as compared to prior deterministic strategies. We also adapt our method to particle-number symmetry, wherein the additional circuit depth may be halved at the cost of roughly 2--5 times more repetitions.