qat.opt.VertexCover

class qat.opt.VertexCover(graph, A=2, B=1, **kwargs)

Specialization of the QUBO class for Vertex Cover.

This class allows for the encoding of a Vertex Cover problem for a given graph and positive constants \(A\) and \(B\). The method produce_q_and_offset() is automatically called. It computes the \(Q\) matrix and QUBO energy offset corresponding to the Hamiltonian representation of the problem, as described in the reference. These are stored in the parent class QUBO and would be needed if one wishes to solve the problem through Simulated Quantum Annealing (SQA) via the SQAQPU - see the Vertex Cover notebook. This QPU also requires a few additional parameters, the specification of which may vary the quality of the solution. We therefore provide the best parameters found thus far through the method get_best_parameters().

For a right encoding, one should ensure that \(A > B\).

Reference

“Ising formulations of many NP problems”, A. Lucas, 2014 - Section 4.3.

import numpy as np
import networkx as nx
from qat.opt import VertexCover

graph = nx.Graph()
graph.add_nodes_from(np.arange(6))
graph.add_edges_from([(0,1), (0,2), (0,3), (0,4), (0,5), (1,5)])
A = 2
B = 1

vertex_cover_problem = VertexCover(graph, A=A, B=B)

print("To anneal the problem, the solver would need "
       + str(len(graph.nodes())) + " spins.")

To anneal the problem, the solver would need 6 spins.
Parameters

  • graph (networkx.Graph) – a networkx graph

  • A (optional, double) – a positive constant by which the terms inside \(H_A\) from \(H = H_A + H_B\) are multiplied, default is 2. This equation comes from the Hamiltonian representation of the problem.

  • B (optional, double) – similar to \(A\), \(B\) is a positive factor for the \(H_B\) terms, default is 1

get_best_parameters()

This method returns a dictionary with the best annealing parameters found thus far after benchmarking. The parameters are needed to produce the entries of the SQAQPU used to solve a Vertex Cover problem via Simulated Quantum Annealing (SQA).

Returns

6-key dictionary containing

  • n_monte_carlo_updates (int) - the number of Monte Carlo updates

  • n_trotters (int) - the number of “classical replicas” or “Trotter replicas”

  • gamma_max (double) - the starting magnetic field

  • gamma_min (double) - the final magnetic field

  • temp_max (double) - the starting temperature

  • temp_min (double) - the final temperature

parse_result(result, inverse=False)

Returns the approximated solution of the Vertex Cover problem from a list of samples

Parameters

result (BatchResult) – BatchResult containing a list of samples

qat.opt.vertex_cover.produce_q_and_offset(graph, A=2, B=1)

Returns the \(Q\) matrix and the offset energy of the problem. The constant \(A\) should be bigger than \(B\) for a right encoding. They are also both positive.

Parameters

  • graph (networkx.Graph) – a networkx graph

  • A (optional, double) – a positive constant by which the terms inside \(H_A\) from \(H = H_A + H_B\) are multiplied, default is 2. This equation comes from the Hamiltonian representation of the problem.

  • B (optional, double) – similar to \(A\), \(B\) is a positive factor for the \(H_B\) terms, default is 1