  Because Majorana zero modes store quantum information non-locally, they are
protected from noise, and have been proposed as a building block for a quantum
computer. We show how to use the same protection from noise to implement
universal fermionic quantum computation. Our architecture requires only two
Majoranas to encode a fermionic quantum degree of freedom, compared to
alternative implementations which require a minimum of four Majoranas for a
spin quantum degree of freedom. The fermionic degrees of freedom support both
unitary coupled cluster variational quantum eigensolver and quantum phase
estimation algorithms, proposed for quantum chemistry simulations. Because we
avoid the Jordan-Wigner transformation, our scheme has a lower overhead for
implementing both of these algorithms, and the simulation of Trotterized
Hubbard Hamiltonian in $\mathcal{O}(1)$ time per unitary step. We finally
demonstrate magic state distillation in our fermionic architecture, giving a
universal set of topologically protected fermionic quantum gates.
