# 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.

Hamiltonian is the basic building block for VQE algorithm. Basically, we try to minimize the expectation of the Hamiltonian to find the lowest eigenvalue of the Hamiltonian matrix which is apparently very useful for many problems ranging from chemistry to finance.

Find more details on: https://arxiv.org/pdf/1907.04769.pdf

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

$$\begin{equation} \text{E}_{ising}(s) = \sum_{i=1}^N h_i s_i + \sum_{i=1}^N \sum_{j=i+1}^N J_{i,j} s_i s_j \end{equation}$$ 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 $$Q$$: $$\begin{equation} 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