# Subtraction¶

Subtraction (denoted as '-') is implemented by negation and addition in 2-complement representation.

$a - b \longleftrightarrow a + (-b) \longleftrightarrow a + \sim{b} + lsb \\_ value$

Where '~' is bitwise not and lsb_value is the least significant bit value.

Note that integer and fixed-point numbers are represented in a 2-complement method during function evaluation. The binary number is extended in the case of a register size miss-match. For example, the positive signed number $$(110)_2=6$$ is expressed as $$(00110)_2$$ when operating with a 5-qubit register. Similarly, the negative signed number $$(110)_2=-2$$ is expressed as $$(11110)_2$$.

Examples:

5 + 3 = 8 , 0101 + 0011 = 1000

5 - 3 = 5 + (-3) = 2, 0101 + 1101 = 0010

-5 + -3 = -8, 1011 + 1101 = 1000

## Syntax¶

Function: Subtractor

Parameters:

• left_arg: Union[float, int, RegisterUserInput] (see RegisterUserInput).
• right_arg: Union[float, int, RegisterUserInput] (see RegisterUserInput).
• output_size: Optional[PositiveInt]
• inplace_arg: Optional[str] = None

Register names:

• left_arg: left_arg
• right_arg: right_arg
• result: difference
{
"function": "Subtractor",
"function_params": {
"left_arg": 3,
"right_arg": { "size": 3 },
"inplace_arg": null
}
}


### Example 1: Two Register¶

{
"functions": [
{
"name": "main",
"body": [
{
"function": "Subtractor",
"function_params": {
"left_arg": {"size": 3},
"right_arg": {"size": 3}
}
}
]
}
]
}

from classiq import Model, QUInt, synthesize
from classiq.builtin_functions import Subtractor

params = Subtractor(
left_arg=QUInt(size=3).to_register_user_input(),
right_arg=QUInt(size=3).to_register_user_input(),
)
model = Model()
model.Subtractor(params)
quantum_program = synthesize(model.get_model())


This example generates a circuit that subtracts two arguments. The left_arg is defined to be a quantum register of size three. The right_arg is defined to be a quantum register of size three.

Output circuit: ### Example 2: Float and Register¶

{
"functions": [
{
"name": "main",
"body": [
{
"function": "Subtractor",
"function_params": {
"left_arg": 3.5,
"right_arg": { "size": 3 }
}
}
]
}
]
}

from classiq import Model, RegisterUserInput, synthesize
from classiq.builtin_functions import Subtractor

params = Subtractor(left_arg=3.5, right_arg=RegisterUserInput(size=3))
model = Model()
model.Subtractor(params)
quantum_program = synthesize(model.get_model())


This example generates a circuit which subtracts two argument. The left_arg is defined to be a fix point number $$(11.1)_2$$. The right_arg is defined to be a quantum register of size of three.

Output circuit: 