Figure 4: Loop $C$ (and antipodal image $\overline{C}$) on the Bloch sphere consisting of geodesic segments that enclose the solid angle. The dynamical phases vanish along each segment. The resulting phases are purely geometric and equal to $-1/2$ for $C$ and $+1/2$ for $\overline{C}$. The computational states 0 and 1 (such as spin-up and spin-down of a spin−1/2 particle) pick up geometric phases of opposite sign, which yield a phase gate fully determined by the solid angle.