Quantum Optimization¶

Introduction to Quantum Optimization¶

Quantum optimization leverages the principles of quantum computing to solve optimization problems more efficiently than classical algorithms. It has the potential to provide significant speedups for various optimization tasks.

Quantum Approximate Optimization Algorithm (QAOA)¶

The Quantum Approximate Optimization Algorithm (QAOA) is a quantum algorithm designed to solve combinatorial optimization problems. It uses a parameterized quantum circuit to approximate the optimal solution.

Example: QAOA for Max-Cut Problem¶

In [ ]:
from qiskit import Aer, transpile, assemble
from qiskit.circuit import Parameter
from qiskit.visualization import plot_histogram
from qiskit.optimization.applications.ising import max_cut
from qiskit.optimization.applications.ising.common import sample_most_likely
from qiskit.optimization.problems import QuadraticProgram
from qiskit.optimization.algorithms import MinimumEigenOptimizer
from qiskit.algorithms import QAOA
from qiskit.utils import QuantumInstance
from qiskit.algorithms.optimizers import COBYLA

# Define the graph for the Max-Cut problem
w = [[0, 1, 1, 0],
     [1, 0, 1, 1],
     [1, 1, 0, 1],
     [0, 1, 1, 0]]

# Convert the problem to a QuadraticProgram
qp = QuadraticProgram()
for i in range(len(w)):
    qp.binary_var(name=f'x{i}')
qp.maximize(linear=[0]*len(w), quadratic=w)

# Convert the problem to an Ising Hamiltonian
qubit_op, offset = max_cut.get_operator(qp)

# Define the QAOA instance
p = 1
qaoa = QAOA(optimizer=COBYLA(), reps=p, quantum_instance=QuantumInstance(Aer.get_backend('qasm_simulator')))

# Solve the problem using QAOA
result = MinimumEigenOptimizer(qaoa).solve(qp)

# Get the solution
solution = sample_most_likely(result.x)
print('Solution:', solution)

# Plot the histogram of the measurement results
counts = result.min_eigen_solver_result.eigenstate
plot_histogram(counts)

Quantum Annealing¶

Quantum annealing is a quantum optimization technique that uses quantum fluctuations to find the global minimum of an optimization problem. It is particularly effective for solving combinatorial optimization problems.

Example: Quantum Annealing for Max-Cut Problem¶

In [ ]:
from qiskit import Aer, transpile, assemble
from qiskit.optimization.applications.ising import max_cut
from qiskit.optimization.applications.ising.common import sample_most_likely
from qiskit.optimization.problems import QuadraticProgram
from qiskit.optimization.algorithms import MinimumEigenOptimizer
from qiskit.algorithms import QAOA
from qiskit.utils import QuantumInstance
from qiskit.algorithms.optimizers import COBYLA

# Define the graph for the Max-Cut problem
w = [[0, 1, 1, 0],
     [1, 0, 1, 1],
     [1, 1, 0, 1],
     [0, 1, 1, 0]]

# Convert the problem to a QuadraticProgram
qp = QuadraticProgram()
for i in range(len(w)):
    qp.binary_var(name=f'x{i}')
qp.maximize(linear=[0]*len(w), quadratic=w)

# Convert the problem to an Ising Hamiltonian
qubit_op, offset = max_cut.get_operator(qp)

# Define the QAOA instance
p = 1
qaoa = QAOA(optimizer=COBYLA(), reps=p, quantum_instance=QuantumInstance(Aer.get_backend('qasm_simulator')))

# Solve the problem using QAOA
result = MinimumEigenOptimizer(qaoa).solve(qp)

# Get the solution
solution = sample_most_likely(result.x)
print('Solution:', solution)

# Plot the histogram of the measurement results
counts = result.min_eigen_solver_result.eigenstate
plot_histogram(counts)

Applications of Quantum Optimization¶

Quantum optimization has significant applications in various fields. Some of the key applications include:

  • Supply Chain Management: Quantum optimization can be used to optimize supply chain operations, including inventory management, transportation, and logistics.
  • Financial Portfolio Optimization: Quantum optimization can be used to optimize financial portfolios, balancing risk and return to achieve the best investment strategy.
  • Scheduling Problems: Quantum optimization can be used to solve complex scheduling problems, such as job scheduling, resource allocation, and project management.
  • Energy Distribution: Quantum optimization can be used to optimize energy distribution in smart grids, reducing energy consumption and improving efficiency.

Conclusion¶

In this notebook, we have explored the fundamental concepts of quantum optimization, including quantum optimization algorithms, example implementations using Qiskit, and their applications. Understanding these concepts is crucial for leveraging quantum computing to solve complex optimization problems more efficiently. As quantum computing technology continues to advance, these techniques will play a key role in revolutionizing various field