Combinatorial structure of a holonomic controlled phase gate

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.