Portfolio Optimization with Python and Quantum Computing Techniques

Portfolio optimization is a fundamental problem in finance where the goal is to find the optimal asset allocation in a portfolio to maximize returns while minimizing risk. Traditional methods of portfolio optimization, such as mean-variance optimization, are based on classical computational methods that can be computationally intensive and do not always produce the best results. Quantum computing, on the other hand, has the potential to revolutionize portfolio optimization by providing a more efficient and accurate solution.

This piece explores how quantum computing can be used to optimize a portfolio and provides examples of Python code that illustrate this process.

Quantum Computing for Portfolio Optimization

Quantum computing is based on the principles of quantum mechanics, which allow quantum computers to perform certain types of computations exponentially faster than classical computers. In the context of portfolio optimization, quantum computing can be used to solve the problem of optimal asset allocation in a portfolio more efficiently and accurately than traditional methods.

The quantum computing approach to portfolio optimization involves encoding the problem as a quantum optimization problem and solving it with a quantum algorithm such as the quantum approximation optimization algorithm (QAOA) or the variational quantum eigenvalue solver (VQE). These algorithms use a series of quantum gates to manipulate qubits, the basic building blocks of quantum computers, to find the optimal solution.

Python Code Examples

To illustrate how quantum computing can be used to optimize a portfolio, we use the Qiskit library, a Python-based open-source framework for quantum computing. We will create a simple portfolio with four assets and use the QAOA algorithm to find the optimal allocation.

First, we will define the portfolio and the expected return and covariance matrix of the assets.

import numpy as np

# Define the portfolio
assets = ['Asset 1', 'Asset 2', 'Asset 3', 'Asset 4']
expected_returns = np.array([0.05, 0.06, 0.07, 0.08])
covariance_matrix = np.array([
    [0.05, 0.02, 0.01, 0.03],
    [0.02, 0.06, 0.02, 0.01],
    [0.01, 0.02, 0.07, 0.03],
    [0.03, 0.01, 0.03, 0.08]
])

Next, we create a quantum optimization by defining the objective function and the constraints. In this case, the objective function is the expected return of the portfolio and the constraints are the total amount of investment and the minimum and maximum allocation for each asset.

from qiskit.optimization import QuadraticProgram
from qiskit.optimization.converters import QuadraticProgramToQubo

# Define the quadratic program
qp = QuadraticProgram('Portfolio Optimization')
qp.binary_var_dict = {asset: 1 for asset in assets}
qp.minimize(np.dot(expected_returns, qp.binary_var_list()) - 0.1 * qp.binary_var_list().dot(covariance_matrix).dot(qp.binary_var_list()))
qp.linear_constraint_dict = {'total_investment': {'lin_expr': {'Asset 1': 1, 'Asset 2': 1, 'Asset 3': 1, 'Asset 4': 1}, 'sense': '==', 'rhs': 1},
                             'min_allocation_1': {'lin_expr': {'Asset 1': 1}, 'sense': '>=', 'rhs': 0.2},
                             'min_allocation_2': {'lin_expr': {'Asset 2': 1}, 'sense': '>=', 'rhs': 0.1},
                             'min_allocation_3': {'lin_expr': {'Asset 3': 1}, 'sense': '>=', 'rhs': 0.1},
                             'min_allocation_4': {'lin_expr': {'Asset 4': 1}, 'sense': '>=', 'rhs': 0.1},
                             'max_allocation_1': {'lin_expr': {'Asset 1': 1}, 'sense': '<=', 'rhs': 0.4},
                             'max_allocation_2': {'lin_expr': {'Asset 2': 1}, 'sense': '<=', 'rhs': 0.3},
                             'max_allocation_3': {'lin_expr': {'Asset 3': 1}, 'sense': '<=', 'rhs': 0.2},
                             'max_allocation_4': {'lin_expr': {'Asset 4': 1}, 'sense': '<=', 'rhs': 0.3}}

Next, we convert the quadratic program to a QUBO, a binary quadratic optimization problem, using the QuadraticProgramToQubo converter.

# Convert the quadratic program to a QUBO
converter = QuadraticProgramToQubo()
qubo = converter.convert(qp)

Finally, we will solve the QUBO using the QAOA algorithm and print the results.

from qiskit.algorithms import QAOA
from qiskit import Aer

# Solve the QUBO using the QAOA algorithm
backend = Aer.get_backend('statevector_simulator')
qaoa = QAOA(reps=1, optimizer=None, quantum_instance=backend)
result = qaoa.compute_minimum_eigenvalue(qubo)
x = qaoa.get_optimal_solution()

# Print the results
print('Optimal solution:', x)
print('Expected return:', np.dot(expected_returns, x))
print('Variance:', x.dot(covariance_matrix).dot(x))

In this example, we used the QAOA algorithm with one repetition to solve the QUBO. The optimal solution and the expected return and variance of the portfolio are output on the console.

Conclusion

In this article, we looked at how quantum computing can be used to optimize a portfolio with Python. We showed how to define a portfolio and constraints, convert a quadratic program to a cube using the QuadraticProgramToQubo converter, and solve QUBO using the QAOA algorithm. Although this example is simple, the quantum computing approach to portfolio optimization has the potential to revolutionize finance by providing more efficient and accurate solutions to complex optimization problems.