# Creating Ising Hamiltonian with Qiskit

How to create Ising Hamiltonian and implement it with qiskit (the output should be in the form of WeightedPauliOperator) for the following problem:

$$H = A \big(K - \sum_{i}^{N} t_i x_i \big)^2\\ x_i=\{0,1\}\\ t_i \in \mathbb{N}_{>0}\\ K\in \mathbb{N}\\ A \in \mathbb{R}_{>0}\\$$

I want to choose any number of variables $$x_i$$, that the sum $$\sum_{i}^{N} t_i x_i$$ will equal exactly $$K$$.

## 1 Answer

You can use Qiskit's new optimization module. This allows you to use docplex to build your model:

# required imports
from docplex.mp.model import Model
from qiskit.optimization.problems import QuadraticProgram
from qiskit.optimization.converters import QuadraticProgramToIsing

# specify problem
n = 3
a = 1.0
k = 2
t = range(1, n+1)

# build model with docplex
mdl = Model()
x = [mdl.binary_var() for i in range(n)]
objective = a*(k - mdl.sum(t[i]*x[i] for i in range(n)))**2
mdl.minimize(objective)

# convert to Qiskit's quadratic program
qp = QuadraticProgram()
qp.from_docplex(mdl)

# convert to Ising Hamiltonian
qp2ising = QuadraticProgramToIsing()
H, offset = qp2ising.encode(qp)
print('Offset:', offset)
print('Ising Hamiltonian:')
print(H.print_details())


Offset: 4.5

Ising Hamiltonian:

IIZ (-1+0j)

IZI (-2+0j)

ZII (-3+0j)

IZZ (1+0j)

ZIZ (1.5+0j)

ZZI (3+0j)

You can find some tutorials introducing the functionality here: https://qiskit.org/documentation/tutorials/optimization/index.html or here https://github.com/Qiskit/qiskit-tutorials/tree/master/tutorials/optimization

• Thanks for the answer! Can you please explain to me one more thing. To run optimization problem (for example with VQE algorithm) it is necessary to get the problem in the form of WeightedPauliOperator. What exactly this structure represent? And how WeightedPauliOperator is related to Ising Hamiltonian? – gosia123 May 21 '20 at 21:07
• The WeightedPauliOperator is a possible way to represent an Ising Hamiltonian, e.g. in contrast to the MatrixOperator. If you want to use VQE or QAOA to actually approximate the ground state, then Qiskit does the translation for you and you can directly use the MinimumEigenOptimizer and give it a QuadraticProgram. See this tutorial on QAOA: github.com/Qiskit/qiskit-tutorials/blob/master/tutorials/… For VQE, just pass VQE as a MinimumEigenSolver instead of QAOA. – Stefan Woerner May 24 '20 at 10:07