  Shadow tomography via classical shadows is a state-of-the-art approach for
estimating properties of a quantum state. We present a simplified,
combinatorial analysis of a recently proposed instantiation of this approach
based on the ensemble of unitaries that are both fermionic Gaussian and
Clifford. Using this analysis, we derive a corrected expression for the
variance of the estimator. We then show how this leads to efficient estimation
protocols for the fidelity with a pure fermionic Gaussian state (provably) and
for an $X$-like operator of the form ($|\mathbf 0\rangle\langle\psi|$ + h.c.)
(via numerical evidence). We also construct much smaller ensembles of
measurement bases that yield the exact same quantum channel, which may help
with compilation. We use these tools to show that an $n$-electron, $m$-mode
Slater determinant can be learned to within $\epsilon$ fidelity given $O(n^2
m^7 \log(m / \delta) / \epsilon^2)$ samples of the Slater determinant.
