  We define and calculate versions of complexity for free fermionic quantum
field theories in 1+1 and 3+1 dimensions, adopting Nielsen's geodesic
perspective in the space of circuits. We do this both by discretizing and
identifying appropriate classes of Bogoliubov-Valatin transformations, and also
directly in the continuum by defining squeezing operators and their
generalizations. As a closely related problem, we consider cMERA tensor
networks for fermions: viewing them as paths in circuit space, we compute their
path lengths. Certain ambiguities that arise in some of these results because
of cut-off dependence are discussed.
