QIRO MaxIndepSet#

Problem description#

Given a Graph \(G = (V,E)\) maximize the size of an independent set, i.e. a subset \(V' \subset V\) in which all pairs of vertices are mutually non-adjacent.

Replacement routine#

Based on the maximum absolute entry of the correlation matrix M and its sign, one of the following replacements is employed:

If \(\text{M}_{ii} \geq 0\) is the maximum absolute value, then the \(i\)-th vertex is set to be in the independent set (IS). In turn, all vertices that share an edge with this vertex can be removed from the graph, since including them in the solution would violate the problem constraints.
If \(\text{M}_{ii} < 0\) is the maximum absolute value, we remove \(i\)-th vertex from the graph.
If \(\text{M}_{ij} > 0, (i, j) ∈ E\) was selected, we remove both vertices from the graph with the argument, that, since both of them would be in the same state in the final solution, including both as part of the solution would violate the constraint, as they share an edge. In turn, they can be removed from the graph.
If \(\text{M}_{ij} < 0, (i, j) ∈ E\) was selected, we remove all vertices that share an edge with both vertices \(i\) and \(j\). Since one of the vertices \(i\) and \(j\) will be part of the final solution (but not both), any vertex that is connected to both \(i\) and \(j\) is guaranteed to violate the problem constraints, and can be removed from the graph. In this case, it may be possible that no vertex is found to be a canditate for being removed. We will then simply choose the second biggest absolute value of M for the replacement routine.

create_max_indep_replacement_routine(res, problem_updated)[source]#

Creates a replacement routine for the problem structure, i.e., defines the replacement rules. See the original paper for a description of the update rules.

Parameters:

resdict: Result dictionary of QAOA optimization procedure.
problem_updatedList: Updates that happened during the QIRO routine. Consits of the updated problem, a list of Qubits which were found to be positively correlated, i.e. part of the problem solution, and a list Qubits which were found to be negatively correlated, i.e. they contradict solution qubits in accordance with the update rules.

Returns:

new_graphnx.Graph: Updated graph for the problem instance.
solutionslist: Updated set of solutions to the problem.
signint: The sign of the correlation.
exclusionslist: Updated set of exclusions for the problem.

QIRO Cost operator#

create_max_indep_cost_operator_reduced(problem_updated)[source]#

Creates the cost_operator for the problem instance. This operator is adjusted to consider qubits that were found to be a part of the problem solution.

Parameters:

problem_updatedList: Updates that happened during the QIRO routine. Consits of the updated problem, a list of Qubits which were found to be positively correlated, i.e. part of the problem solution, and a list Qubits which were found to be negatively correlated, i.e. they contradict solution qubits in accordance with the update rules.

Returns:

cost_operatorfunction: A function receiving a QuantumVariable and a real parameter \(\gamma\). This function performs the application of the cost operator.

QIRO with Constrained mixer#

create_max_indep_controlled_mixer_reduced(problem_updated)[source]#

Creates the controlled_RX_mixer for a QIRO instance of the maximal independet set problem for a given graph G following Hadfield et al.

The belonging predicate function indicates if a set can be swapped into the solution.

Parameters:

problem_updatedList: Updates that happened during the QIRO routine. Consits of the updated problem, a list of Qubits which were found to be positively correlated, i.e. part of the problem solution, and a list Qubits which were found to be negatively correlated, i.e. they contradict solution qubits in accordance with the update rules.

Returns:

controlled_RX_mixerfunction: A Python function receiving a QuantumVariable and real parameter \(\beta\). This function performs the application of the mixer associated to the graph G.

qiro_max_indep_set_init_function(solutions=[], exclusions=[])[source]#

To be used for the controlled mixer approach of QIRO MIS. Only flips qubits which we found to be a part of the problem soultion.

Parameters:

solutionsList: List of Qubits which were found to be positively correlated, i.e. part of the problem solution
exclusionsList: List Qubits which were found to be negatively correlated, i.e. they contradict solution qubits in accordance with the update rules.

Example implementation#

The implementation below is for the standard unconstrained cost operator implementation. An implementation with a controlled mixer ansatz is shown in the tutorial section.

from qrisp import QuantumVariable
from qrisp.qiro import QIROProblem, create_max_indep_replacement_routine, create_max_indep_cost_operator_reduced, qiro_rx_mixer, qiro_init_function
from qrisp.qaoa import create_max_indep_set_cl_cost_function
import matplotlib.pyplot as plt
import networkx as nx

# Define a random graph via the number of nodes and the QuantumVariable arguments
num_nodes = 13
G = nx.erdos_renyi_graph(num_nodes, 0.4, seed = 107)
qarg = QuantumVariable(G.number_of_nodes())

qiro_instance = QIROProblem(G,
                            replacement_routine=create_max_indep_replacement_routine,
                            cost_operator=create_max_indep_cost_operator_reduced,
                            mixer=qiro_rx_mixer,
                            cl_cost_function=create_max_indep_set_cl_cost_function,
                            init_function=qiro_init_function
                            )
res_qiro = qiro_instance.run_qiro(qarg=qarg, depth=3, n_recursions=2)

That’s it! In the following, we print the 5 most likely solutions together with their cost values, and compare to the NetworkX solution.

cl_cost = create_max_indep_set_cl_cost_function(G)

print("5 most likely QIRO solutions")
max_five_qiro = sorted(res_qiro, key=res_qiro.get, reverse=True)[:5]
for res in max_five_qiro:
    print([index for index, value in enumerate(res) if value == '1'])
    print(cl_cost({res : 1}))

print("Networkx solution")
print(nx.approximation.maximum_independent_set(G))

Finally, we visualize the final QIRO graph and the most likely solution.

# The final graph that has been adjusted
final_graph = qiro_instance.problem

# Draw the final graph and the original graph for comparison
plt.figure(1)
nx.draw(final_graph, with_labels=True, node_color='#ADD8E6', edge_color='#D3D3D3')
plt.title('Final QIRO graph')

plt.figure(2)
most_likely = [index for index, value in enumerate(max_five_qiro[0]) if value == '1']
nx.draw(G, with_labels=True,
        node_color=['#FFCCCB' if node in most_likely else '#ADD8E6' for node in G.nodes()],
        edge_color='#D3D3D3')
plt.title('Original graph with most likely QIRO solution')
plt.show()