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 $\binom{\eta}{k}\big(1-\frac{\eta-k}{n}\big)^{k}\frac{1+n}{1+n-k}/\epsilon^{2}$ 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 $\mathcal{O}(\binom{n}{k}\frac{\sqrt{k}}{\epsilon^{2}})$ 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 $\eta$.