CX gates are ubiquitous in quantum algorithms, making the minimization of CX gates an important optimization problem. CX gates generate a group isomorphic to GL(n,F_2), the group of invertible matrices over the field of two elements F_2 = {0,1}.
This report investigates a possible connection between CX minimization and the theory of the spherical building associated to the group GL(n,F_2). We formulate two questions directly related to CX minimization. By looking at particular subsets of the generating CX gates, we obtain two related but more tractable questions.
We answer the first of these two questions and give an outline of a solution to the second. The report also serves as a self-contained introduction to CX minimization and the theory of buildings.
