Can we use fermionic modes as qubits? They both are two state systems, the only difference being the exchange statistics.
 A: There is a classic work by Bravyi and Kitaev that shows how one can embed $m$ number of qubits into fermionic modes such that any unitary acting on the $m$ quits can be represented in an easy way as a unitary on the fermionic system. Here is a link: https://arxiv.org/pdf/quant-ph/0003137.pdf . (And also the parity super-selection rule is respected in this scheme.)
A: Sure, that might be hard to implement in practice, but in principle any two-level system works.
One issue would be that it's difficult to physically implement operators that violate fermion parity, so "$\sigma^x$" gates like $a + a^\dagger$ would be difficult to implement. But you could in principle implement them in an open system by hopping fermions in and out out of the system.
Another issue is that if you had multiple qubits, then you'd probably want to keep them well-separated, so that you could neglect the antisymmetrization requirement and effectively work with the full tensor-product Hilbert space. In this case it could be quite tricky to entangle the faraway qubits together in practice.
