  Recently multiple families of spin chain models were found, which have a free
fermionic spectrum, even though they are not solvable by a Jordan-Wigner
transformation. Instead, the free fermions emerge as a result of a rather
intricate construction. In this work we consider the quantum circuit
formulation of the problem. We construct circuits using local unitary gates
built from the terms in the local Hamiltonians of the respective models, and
ask the question: which circuit geometries (sequence of gates) lead to a free
fermionic spectrum? Our main example is the 4-fermion model of Fendley, where
we construct free fermionic circuits with various geometries. In certain cases
we prove the free fermionic nature, while for other geometries we confirm it
numerically. Surprisingly, we find that many standard brickwork circuits are
not free fermionic, but we identify certain symmetric constructions which are.
