  We study brick wall quantum circuits enjoying a global fermionic symmetry.
The constituent 2-qubit gate, and its fermionic symmetry, derive from a
2-particle scattering matrix in integrable, supersymmetric quantum field theory
in 1+1 dimensions. Our 2-qubit gate, as a function of three free parameters, is
of so-called free fermionic or matchgate form, allowing us to derive the
spectral structure of both the brick wall unitary $U_F$ and its, non-trivial,
hamiltonian limit $H_{\gamma}$ in closed form. We find that the fermionic
symmetry pins $H_{\gamma}$ to a surface of critical points, whereas breaking
that symmetry leads to non-trivial topological phases. We briefly explore
quench dynamics for this class of circuits.
