Fermionic partial tomography via classical shadows

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 $\epsilon$ requires on the order of $\left(\genfrac{}{}{0ex}{}{n}{k}\right)k^{3/2}\log (n)/{\epsilon }^2$ 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 $k\ge 2$, 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.

5+26 pages, 5 figures, 1 table. The SM has been revised to replace an incorrect variance expression for arbitrary observables with an upper bound instead. The results presented in the main text are unaffected by this revision

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