Creating Ising Hamiltonian with Qiskit

Question

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 $K$.

Answer 1

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