  An important class of fermionic observables, relevant in tasks such as
fermionic partial tomography and estimating energy levels of chemical
Hamiltonians, are the binary measurements obtained from the product of
anti-commuting Majorana operators. In this work, we investigate efficient
estimation strategies of these observables based on a joint measurement which,
after classical post-processing, yields all sufficiently unsharp (noisy)
Majorana observables of even-degree. By exploiting the symmetry properties of
the Majorana observables, as described by the braid group, we show that the
incompatibility robustness, i.e., the minimal classical noise necessary for
joint measurability, relates to the spectral properties of the
Sachdev-Ye-Kitaev (SYK) model. In particular, we show that for an $n$ mode
fermionic system, the incompatibility robustness of all degree--$2k$ Majorana
observables satisfies $\Theta(n^{-k/2})$ for $k\leq 5$. Furthermore, we present
a joint measurement scheme achieving the asymptotically optimal noise,
implemented by a small number of fermionic Gaussian unitaries and sampling from
the set of all Majorana monomials. Our joint measurement, which can be
performed via a randomization over projective measurements, provides rigorous
performance guarantees for estimating fermionic observables comparable with
fermionic classical shadows.
