def test_clique(self):
N = 5
graph = retworkx.PyGraph()
graph.extend_from_edge_list([(i, j) for i in range(N) for j in range(N)
if i < j])
complement_graph = retworkx.complement(graph)
self.assertEqual(graph.nodes(), complement_graph.nodes())
self.assertEqual(0, len(complement_graph.edges()))
def test_multigraph(self):
graph = retworkx.PyGraph(multigraph=True)
graph.extend_from_edge_list([(0, 0), (0, 1), (1, 1), (2, 2), (1, 0)])
expected_graph = retworkx.PyGraph(multigraph=True)
expected_graph.extend_from_edge_list([(0, 2), (1, 2)])
complement_graph = retworkx.complement(graph)
self.assertTrue(
retworkx.is_isomorphic(
expected_graph,
complement_graph,
))
def test_empty(self):
N = 5
graph = retworkx.PyGraph()
graph.add_nodes_from([i for i in range(N)])
expected_graph = retworkx.PyGraph()
expected_graph.extend_from_edge_list([(i, j) for i in range(N)
for j in range(N) if i < j])
complement_graph = retworkx.complement(graph)
self.assertTrue(
retworkx.is_isomorphic(
expected_graph,
complement_graph,
))
def test_complement(self):
N = 8
graph = retworkx.PyGraph()
graph.extend_from_edge_list([(j, i) for i in range(N) for j in range(N)
if i < j and (i + j) % 3 == 0])
expected_graph = retworkx.PyGraph()
expected_graph.extend_from_edge_list([(i, j) for i in range(N)
for j in range(N)
if i < j and (i + j) % 3 != 0])
complement_graph = retworkx.complement(graph)
self.assertTrue(
retworkx.is_isomorphic(
expected_graph,
complement_graph,
))
def test_null_graph(self):
graph = retworkx.PyGraph()
complement_graph = retworkx.complement(graph)
self.assertEqual(0, len(complement_graph.nodes()))
self.assertEqual(0, len(complement_graph.edges()))
def max_clique(graph: Union[nx.Graph, rx.PyGraph], constrained: bool = True):
r"""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 <https://en.wikipedia.org/wiki/Clique_(graph_theory)>`__ of a
graph --- the largest subgraph such that all vertices are connected by an edge.
Args:
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:
(.Hamiltonian, .Hamiltonian): The cost and mixer Hamiltonians
.. UsageDetails::
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:
.. math:: H_C \ = \ \displaystyle\sum_{v \in V(G)} Z_{v},
where :math:`V(G)` is the set of vertices of the input graph, and :math:`Z_i` is the Pauli-Z operator
applied to the :math:`i`-th
vertex.
The returned mixer Hamiltonian is :func:`~qaoa.bit_flip_mixer` applied to :math:`\bar{G}`,
the complement of the graph.
.. note::
**Recommended initialization circuit:**
Each wire in the :math:`|0\rangle` state.
**Unconstrained**
The Maximum Clique cost Hamiltonian for unconstrained QAOA is defined as:
.. math:: 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 :math:`V(G)` is the set of vertices of the input graph :math:`G`, :math:`E(\bar{G})` is the set of
edges of the complement of :math:`G`, and :math:`Z_i` is the Pauli-Z operator applied to the
:math:`i`-th vertex.
The returned mixer Hamiltonian is :func:`~qaoa.x_mixer` applied to all wires.
.. note::
**Recommended initialization circuit:**
Even superposition over all basis states.
"""
if not isinstance(graph, (nx.Graph, rx.PyGraph)):
raise ValueError(
f"Input graph must be a nx.Graph or rx.PyGraph, got {type(graph).__name__}"
)
graph_nodes = graph.nodes()
graph_complement = (rx.complement(graph) if isinstance(graph, rx.PyGraph)
else nx.complement(graph))
if constrained:
cost_h = bit_driver(graph_nodes, 1)
cost_h.grouping_indices = [list(range(len(cost_h.ops)))]
return (cost_h, qaoa.bit_flip_mixer(graph_complement, 0))
cost_h = 3 * edge_driver(graph_complement,
["10", "01", "00"]) + bit_driver(graph_nodes, 1)
mixer_h = qaoa.x_mixer(graph_nodes)
# store the valuable information that all observables are in one commuting group
cost_h.grouping_indices = [list(range(len(cost_h.ops)))]
return (cost_h, mixer_h)
