Complexity of graph-state preparation by Clifford circuits

In this work, we study the complexity of graph-state preparation in a general model of quantum algorithms that allows measurements in the computational basis, single-qubit Clifford operations, and two-qubit Clifford operations. We define the CZ-complexity of a graph state | G ⟩ as the minimum number of two-qubit Clifford operations required to generate | G ⟩ from | 0 ⟩ ⊗ ( n + s ) for some s ≥ 0 . Equivalently, every optimal algorithm can be taken to use only controlled-Z (CZ) gates as its two-qubit Clifford operations. We then give a combinatorial characterization of graph-state transformations. Specifically, | G ⟩ can be generated from another graph state | H ⟩ by an algorithm of CZ-complexity at most t if and only if G can be obtained from H by vertex deletions, local complementations and at most t elementary edge-complementations. Here, an elementary edge-complementation toggles either a single edge, all edges between one vertex and the neighborhood of another, or all edges between the neighborhoods of two non-adjacent vertices. Using this characterization, we relate CZ-complexity to rank-width. For any graph G with n vertices and rank-width r , the CZ-complexity is O ( r n ) , and if G is connected then it is at least n + r − 2 . We also show that these bounds are close to optimal. Finally, for interval graphs and circle graphs, whose rank-width is unbounded, we present preparation algorithms with CZ-complexity O ( n ) and O ( n log ⁡ n ) , respectively.