  We present a general construction of a geometric notion of circuit complexity
for Gaussian states (both bosonic and fermionic) in terms of Riemannian
geometry. We lay out general conditions that a Riemannian metric function on
the space of Gaussian states should satisfy in order for it to yield a
physically reasonable measure of complexity. This general formalism can
naturally accommodate modifications to complexity geometries that arise from
cost functions that depend nontrivially on the instantaneous state and on the
direction on circuit space at each point. We explore these modifications and,
as a particular case, we show how to account for time-reversal symmetry
breaking in measures of complexity, which is often natural from an experimental
(and thermodynamical) perspective, but is absent in commonly studied complexity
measures. This establishes a first step towards building a quantitative,
geometric notion of complexity that faithfully mimics what is experienced as
"easy" or "hard" to implement in a lab from a physically motivated point of
view.

Quantum Physics;General Relativity and Quantum Cosmology;High Energy Physics - Theory