# Ising¶

```minimize  - x^T J x - h^T x + C
x_i is in {-1, 1}
```

This optimization problem is called Ising model.

## flopt to Ising¶

For example, the following problem is one of the Ising using Spin variables.

```from flopt import Variable, Problem

# Variables
a = Variable("a", cat="Spin")
b = Variable("b", cat="Spin")

# Problem
prob = Problem()
prob += 1 - a * b - a

print(prob)
>>> Name: None
>>>   Type         : Problem
>>>   sense        : minimize
>>>   objective    : 1-a*b-a
>>>   #constraints : 0
>>>   #variables   : 2 (Spin 2)
```

We can convert this into Ising form as follows.

```from flopt.convert import IsingStructure
ising = IsingStructure.fromFlopt(prob)
```

To show the contents of ising structure,

```print(ising.show())
>>> IsingStructure
>>> - x.T.dot(J).dot(x) - h.T.dot(x) + C
>>>
>>> #x
>>> 2
>>>
>>> J
>>> [[-0.  1.]
>>>  [-0. -0.]]
>>>
>>> h
>>> [ 1. -0.]
>>>
>>> C
>>> 1
>>>
>>> x
>>> [Variable("a", cat="Spin", ini_value=1)
>>>  Variable("b", cat="Spin", ini_value=1)]
```

We can convert flopt to ising even if the problem includes binary variable. Binary variables are automatically replaced to spin variable.

```from flopt import Variable, Problem

# Variables
a = Variable("a", cat="Spin")
b = Variable("b", cat="Binary") # Binary variable

# Problem
prob = Problem()
prob += 1 - a * b - a

print(prob)
>>> Name: None
>>>   Type         : Problem
>>>   sense        : minimize
>>>   objective    : 1-(a*b)-a
>>>   #constraints : 0
>>>   #variables   : 2 (Binary 1, Spin 1)

from flopt.convert import IsingStructure
ising = IsingStructure.fromFlopt(prob)

print(ising.show())
>>> IsingStructure
>>> - x.T.dot(J).dot(x) - h.T.dot(x) + C
>>>
>>> #x
>>> 2
>>>
>>> J
>>> [[-0.   0.5]
>>>  [-0.  -0. ]]
>>>
>>> h
>>> [ 1.5 -0. ]
>>>
>>> C
>>> 1.0
>>>
>>> x
>>> [Variable("a", cat="Spin", ini_value=-1)
>>>  Variable("b_s", cat="Spin", ini_value=-1)]
```

b_s is the spin variable as b_s = 2 b - 1.

### Convert to QUBO¶

To convert this problem as QUBO formulation, we use .toQubo() function.

```ising.toQubo()    # convert ising to QUBO

print(ising.toQubo().toFlopt().show())  # for show cleary ising.toQubo()
>>> Name: None
>>>   Type         : Problem
>>>   sense        : minimize
>>>   objective    : -4.0*(a_b*b_b)+(2.0*b_b)+1.0
>>>   #constraints : 0
>>>   #variables   : 2 (Binary 2)
```

a_b is the binary variable as a_b = (1+a)/2. In addition, .toQp(), .toLp() are also available.

## Ising to flopt¶

```# make ising model
J = [[0, 1],
[0, 0]]
h = [1, 0]
C = 1

from flopt.convert import IsingStructure
prob = IsingStructure(J, h, C).toFlopt()

prob.show()
>>> Name: None
>>>   Type         : Problem
>>>   sense        : minimize
>>>   objective    : -x_0*x_1-x_0+1
>>>   #constraints : 0
>>>   #variables   : 2 (Spin 2)
```