Fermionic Partial Tomography via Classical Shadows

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 $(\genfrac{}{}{0ex}{}{n}{k})k^{3/2}\log (n)/{ϵ}^2$ 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 $k\ge 2$, 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.

Figure 1

(Left) Summary of the methods compared here, cataloging their required circuit types and scalings in the number of measurement settings. Because graph-based methods [33, 34, 35, 38] are resource intensive, we employ sorted insertion [17] as a more tractable alternative. The Majorana clique cover [37], which employs the same class of fermionic Gaussian Clifford circuits as our classical shadows (FGU) unitaries, possesses optimal asymptotic scaling; however, it exhibits jumps at powers of 2 due to a divide-and-conquer approach. Furthermore, the construction exists only for $k\le 2$. The measurement strategy using fermionic swap networks is a generalization of the optimal 1-RDM strategy introduced in Ref. [61], which we describe in Sec. IV of the Supplemental Material [83]. (Right) Numerical performances (log-log scale). Note that sorted insertion and the Majorana clique cover are equivalent for $k=1$. Because our scheme uses randomization, we include error bars of 1 standard deviation, averaged over 10 instances. However, they are not visible at the scale of the plots, indicating the consistency of our method.