  We propose a simple scheme to estimate fermionic observables and Hamiltonians
relevant in quantum chemistry and correlated fermionic systems. Our approach is
based on implementing a measurement that jointly measures noisy versions of any
product of two or four Majorana operators in an $N$ mode fermionic system. To
realize our measurement we use: (i) a randomization over a set of unitaries
that realize products of Majorana fermion operators; (ii) a unitary, sampled at
random from a constant-size set of suitably chosen fermionic Gaussian
unitaries; (iii) a measurement of fermionic occupation numbers; (iv) suitable
post-processing. Our scheme can estimate expectation values of all quadratic
and quartic Majorana monomials to $\epsilon$ precision using $\mathcal{O}(N
\log(N)/\epsilon^2)$ and $\mathcal{O}(N^2 \log(N)/\epsilon^2)$ measurement
rounds respectively, matching the performance offered by fermionic shadow
tomography. In certain settings, such as a rectangular lattice of qubits which
encode an $N$ mode fermionic system via the Jordan-Wigner transformation, our
scheme can be implemented in circuit depth $\mathcal{O}(N^{1/2})$ with
$\mathcal{O}(N^{3/2})$ two-qubit gates, offering an improvement over fermionic
and matchgate classical shadows that require depth $\mathcal{O}(N)$ and
$\mathcal{O}(N^2)$ two-qubit gates. By benchmarking our method on exemplary
molecular Hamiltonians and observing performances comparable to fermionic
classical shadows, we demonstrate a novel, competitive alternative to existing
strategies.
