  We investigate the combinatorial structures of a holonomic controlled quantum
gate based on toric varieties. In particular, we in detail discuss the
combinatorial structures of a two-qubit holonomic controlled quantum gate on a
two-qubit, a three-qubit, and a four-qubit quantum states. Our results show
interesting relations between toric varieties of qubit systems and action of a
two-qubit holonomic quantum gate on multi-qubit states.
