Classical shadows of fermions with particle number symmetry

Abstract

We consider classical shadows of fermion wavefunctions with $\eta$ particles occupying $n$ modes. We prove that all $k$-Reduced Density Matrices (RDMs) may be simultaneously estimated to an average variance of ${\epsilon }^2$ using at most $\left(\genfrac{}{}{0ex}{}{\eta }{k}\right)(1-\frac{\eta -k}{n}{)}^k\frac{1+n}{1+n-k}/{\epsilon }^2$ measurements in random single-particle bases that conserve particle number, and provide an estimator for any $k$-RDM with $\mathcal{O}(k^2\eta )$ classical complexity. Our sample complexity is a super-exponential improvement over the $\mathcal{O}((\genfrac{}{}{0ex}{}{n}{k})\frac{\sqrt{k}}{{\epsilon }^2})$ scaling of prior approaches as $n$ can be arbitrarily larger than $\eta$, which is common in natural problems. Our method, in the worst-case of half-filling, still provides a factor of $4^k$ advantage in sample complexity, and also estimates all $\eta$-reduced density matrices, applicable to estimating overlaps with all single Slater determinants, with at most $\mathcal{O}(\frac{1}{{\epsilon }^2})$ samples, which is additionally independent of $\eta$.

26 pages. V2: Faster estimator

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