(this question is a crosspost from cstheory. I've incorporated the one answer there into the question)
Some of us have been reading Michael Nielsen's paper on a geometric approach to using quantum lower bounds (in brief, the construction of a Finsler metric on $SU(2^n)$ such that the geodesic distance from $I$ to an element $U$ is a lower bound on the number of gates in a quantum circuit that computes $U$).
I was wondering if there were concrete examples of problems where this program led to a lower bound that came close to, matched or beat prior lower bounds obtained by other means ?
One example that Joe Fitzsimmons provided is this paper on optimal transfer rates in spin chains. While it's a good example of the "spirit" of the original idea, I'm specifically looking for methods that use Nielsen's program to provide lower bounds.