qml.qaoa.cost.max_independent_set

max_independent_set(graph, constrained=True)[source]

For a given graph, returns the QAOA cost Hamiltonian and the recommended mixer corresponding to the Maximum Independent Set problem.

Given some graph \(G\), an independent set is a set of vertices such that no pair of vertices in the set share a common edge. The Maximum Independent Set problem, is the problem of finding the largest such set.

Parameters
  • graph (nx.Graph) – 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

(Hamiltonian, Hamiltonian)

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 Independent Set 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 Pauli-Z operator applied to the \(i\)-th vertex.

The returned mixer Hamiltonian is bit_flip_mixer() applied to \(G\).

Note

Recommended initialization circuit:

Each wire in the \(|0\rangle\) state.

Unconstrained

The Maximum Independent Set cost Hamiltonian for unconstrained QAOA is defined as:

\[H_C \ = \ 3 \sum_{(i, j) \in E(G)} (Z_i Z_j \ - \ Z_i \ - \ Z_j) \ + \ \displaystyle\sum_{i \in V(G)} Z_i\]

where \(E(G)\) is the set of edges of \(G\), \(V(G)\) is the set of vertices, and \(Z_i\) is the Pauli-Z operator acting on 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.