  We study circuit complexity for free fermionic field theories and Gaussian
states. Our definition of circuit complexity is based on the notion of geodesic
distance on the Lie group of special orthogonal transformations equipped with a
right-invariant metric. After analyzing the differences and similarities to
bosonic circuit complexity, we develop a comprehensive mathematical framework
to compute circuit complexity between arbitrary fermionic Gaussian states. We
apply this framework to the free Dirac field in four dimensions where we
compute the circuit complexity of the Dirac ground state with respect to
several classes of spatially unentangled reference states. Moreover, we show
that our methods can also be applied to compute the complexity of excited
states. Finally, we discuss the relation of our results to alternative
approaches based on the Fubini-Study metric, the relevance to holography and
possible extensions.
