Complexity of graph-state preparation by Clifford circuits

In this work, we study a complexity of graph-state preparation. We consider general quantum algorithms consisting of the Clifford operations on at most two qubits for graph-state preparations. We define the CZ-complexity of graph state |G⟩ as the minimum number of two-qubit Clifford operations (excluding single-qubit Clifford operations) for generating |G⟩ from a trivial state |0⟩^⊗ n. We first prove that a graph state |G⟩ is generated by at most t two-qubit Clifford operations if and only if |G⟩ is generated by at most t controlled-Z (CZ) operations. We next prove that a graph state |G⟩ is generated from another graph state |H⟩ by t CZ operations if and only if the graph G is generated from H by some combinatorial graph transformation with cost t. As the main results, we show a connection between the CZ-complexity of graph state |G⟩ and the rank-width of the graph G. Indeed, we prove that for any graph G with n vertices and rank-width r, 1. The CZ-complexity of |G⟩ is O(rnlog n). 2. If G is connected, the CZ-complexity of |G⟩ is at least n + r - 2. We also show the existence of graph states whose CZ-complexities are close to the upper and lower bounds. Finally, we present quantum algorithms preparing |G⟩ with O(n) CZ-complexity when G is included in special classes of graphs, namely, cographs, interval graphs, permutation graphs and circle graphs.