Toward a physically motivated notion of Gaussian complexity geometry

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.