How to formulate Portfolio Optimization problems with quantum algorithms?

Can anyone help explain how to formulate a portfolio optimization problem into a quantum algorithm?

Problem setup: Similar to the simple financial case, assume that we have a list of investment options, and know the corresponding covariance matrix $\Sigma$ and expected return $\mu$. I have budget $B$ and I can accept risks quantified by a risk factor $q$. I need to decide which options to buy within my budget, such that I can maximize the returns but minimize the risks.

Questions:
1. How can I use quantum algorithms to find the optimal portfolio?
2. What is the advantage of this quantum algorithm compared to classical algorithms?

Thank you very much!

Let me try to answer the first question about how to use quantum algorithms to optimize the portfolio.

Step 1. Formulate the problem in a quadratic form
Step 2. Find the Hamiltonian corresponds to the quadratic problem
Step 3. Choose a method to search for the ground state of the Hamiltonian, which would be the optimal solution

Step 1

Let x be the vector that record the asset allocation. The optimal portfolio maximize the return and minimize the risk, which is equivalent to minimizing
$$
q x^T \Sigma x - \mu^T x,
$$
which is subject to
$$
\mathbb{1}^T x = B
$$

This is a typical quadratic optimization problem. In Qiskit, you can code it simple by calling the PortfolioOptimization package

from qiskit_finance.applications.optimization import PortfolioOptimization

# Define the portfolio
portfolio = PortfolioOptimization(expected_returns = mu, covariances = sigma, risk_factor = q, budget = B, bounds = None)

# Translate the portfolio into a quadratic program
qp = portfolio.to_quadratic_program()

Step 2 and 3

In quantum computation, a quadratic optimization problem can be solved as a ground state search of a Hamiltonian. Especially, in simple cases where elements of $x$ can only take 1 or 0 (in other words, you only decide whether to allocate investment in a certain asset), the corresponding quantum model is called Ising model, where every element in $x$ is mapped to a particle with spin up or spin down.

Qiskit has very well written package to help you do this mapping from quadratic problem to Hamiltonian. You only need to choose a method to search for the ground state, such as VQE.

from qiskit import Aer
from qiskit.algorithms import VQE
from qiskit.algorithms.optimizers import *

optimizer = SLSQP(maxiter=1000)  
algorithm_globals.random_seed = 1234
backend = Aer.get_backend('statevector_simulator')

vqe = VQE(optimizer=optimizer, quantum_instance=backend)

vqe_meo = MinimumEigenOptimizer(vqe)  

result = vqe_meo.solve(qp)