qml.qaoa.cost.max_clique¶

max_clique
(graph: Union[networkx.classes.graph.Graph, retworkx.PyGraph], constrained: bool = True)[source]¶ Returns the QAOA cost Hamiltonian and the recommended mixer corresponding to the Maximum Clique problem, for a given graph.
The goal of Maximum Clique is to find the largest clique of a graph — the largest subgraph such that all vertices are connected by an edge.
 Parameters
graph (nx.Graph or rx.PyGraph) – a graph whose edges define the pairs of vertices on which each term of the Hamiltonian acts
constrained (bool) – specifies the variant of QAOA that is performed (constrained or unconstrained)
 Returns
The cost and mixer Hamiltonians
 Return type
Usage Details
There are two variations of QAOA for this problem, constrained and unconstrained:
Constrained
Note
This method of constrained QAOA was introduced by Hadfield, Wang, Gorman, Rieffel, Venturelli, and Biswas in arXiv:1709.03489.
The Maximum Clique cost Hamiltonian for constrained QAOA is defined as:
\[H_C \ = \ \displaystyle\sum_{v \in V(G)} Z_{v},\]where \(V(G)\) is the set of vertices of the input graph, and \(Z_i\) is the PauliZ operator applied to the \(i\)th vertex.
The returned mixer Hamiltonian is
bit_flip_mixer()
applied to \(\bar{G}\), the complement of the graph.Note
 Recommended initialization circuit:
Each wire in the \(0\rangle\) state.
Unconstrained
The Maximum Clique cost Hamiltonian for unconstrained QAOA is defined as:
\[H_C \ = \ 3 \sum_{(i, j) \in E(\bar{G})} (Z_i Z_j \  \ Z_i \  \ Z_j) \ + \ \displaystyle\sum_{i \in V(G)} Z_i\]where \(V(G)\) is the set of vertices of the input graph \(G\), \(E(\bar{G})\) is the set of edges of the complement of \(G\), and \(Z_i\) is the PauliZ operator applied to the \(i\)th vertex.
The returned mixer Hamiltonian is
x_mixer()
applied to all wires.Note
 Recommended initialization circuit:
Even superposition over all basis states.