Figure 4:
Loop C (and antipodal image C̅ ) 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 C̅ . 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.