as [@KW15] which we use here. The reason to use [@KW15], other than efficiency, is because the construction is “local” and the server can obtain some information about the true graph (needed for 2PQC) without compromising the security. The construction of the resource given a base-graph $G$ (graph that the UBQC computation without traps requires), that has vertices $v\in V(G)$ and edges $e\in E(G)$, is the following:

1.  For each vertex $v_i$, we define a set of three new vertices $P_{v_i}=\{p^{v_i}_1,p^{v_i}_3,p^{v_i}_3\}$. These are called *primary* vertices.

2.  Corresponding to each edge $e(v_i,v_j)\in E(G)$ of the base-graph that connects the base vertices $v_i$ and $v_j$, we introduce a set of nine edges $E_{e(v_i,v_j)}$ that connect each of the vertices in the set $P_{v_i}$ with each of the vertices in the set $P_{v_j}$.

3.  We replace every edge in the resulting graph with a new vertex connected to the two vertices originally joined by that edge. The new vertices added in this step are called *added* vertices. This is the *dotted triple-graph* $DT(G)$.

The edge or vertex of the initial graph that each vertex $v\in DT(G)$ belongs to is called its *base-location*. We can see that by inserting dummy qubits among the 