What is Ising Hamiltonian ? What its role in Portfolio Diversification?

I am asking this question with reference to this https://github.com/Qiskit/qiskit-iqx-tutorials/blob/master/qiskit/advanced/aqua/finance/optimization/portfolio_diversification.ipynb Happy to know new resources on portfolio optimization.

1 Answer

The Ising model is a formulation of your problem. Variables are variables $$s_i$$ that can take +1/-1 values.

$$$$\text{E}_{ising}(s) = \sum_{i=1}^N h_i s_i + \sum_{i=1}^N \sum_{j=i+1}^N J_{i,j} s_i s_j$$$$ For a quantum form, we use spin operators $$\sigma^z$$, giving you an Ising Hamiltonian, whose eigenvalues correspond to the previous cost. What we try to achieve for minimization, is find the state $$s$$ minimizing the Ising model. Which corresponds to finding the minimal eigenvalue or ground state of the Hamiltonian. And this is what research/applications in combinatorial optimization using quantum computers are focusing on.

In computer science, there is a similar formulation. We use generally QUBO formulations defined on 0/1 variables $$x_i$$ with a matrix of coefficients $$Q$$: $$$$f(x) = \sum_{i} {Q_{i,i}}{x_i} + \sum_{i

It is very easy to go from one formulation to another, using the transformation $$s = 2x - 1$$. Many problems can be expressed with these two formulations such as Portfolio Optimization.

For references, you can have a look at (in addition to the notebook you point out):

Ising formulations of many NP problems

D-Wave documentation on Ising/QUBO