# Multiplication¶

The "Multiplication" operation, denoted '$$*$$', can be seen as a series of addition ("long multiplication"). Therefore, there are different implementations to the multiplier depending on the of adder being used.

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¶

The calculation of -5 * 3 = -15 is done in the following manner:

The left arg -5 is represented as 1011 and 3 as 11. The number of digits needed to store the answer would be 4+2-1 = 5. The multiplication is done in the 'regular' way where each number is extended to 5 bits and only 5 digit are kept in the intermediary results:

$\begin{equation*}\begin{array}{c} \phantom{\times}11011\\ \underline{\times\phantom{000}11}\\ \phantom{\times}11011\\ \underline{\phantom\times1011\phantom9}\\ \phantom\times10001 \end{array}\end{equation*}$

## Syntax¶

Function: Multiplier

Parameters:

{
"function": "Multiplier",
"function_params": {
"left_arg": 3,
"right_arg": {
"size": 3
},
"inplace": false
}
}


### Example 1: Two Register Multiplication¶

{
"constraints": {
"max_width": 20,
"max_depth": 300
},
"logic_flow": [
{
"function": "Multiplier",
"function_params": {
"left_arg": {"size": 3},
"right_arg": {"size": 3}
}
}
]
}

from classiq import ModelDesigner, QUInt
from classiq.builtin_functions import Multiplier

params = Multiplier(
left_arg=QUInt(size=3).to_register_user_input(),
right_arg=QUInt(size=3).to_register_user_input(),
)
model_designer = ModelDesigner()
model_designer.Multiplier(params)
circuit = model_designer.synthesize()
circuit.show()


This code example generates a circuit that multiplies 2 arguments. Both "left_arg" and "right_arg" are defined to be quantum registers of size 3.

#### Generated Circuit¶ ### Example 2: Float and Register Multiplication¶

{
"constraints": {
"max_width": 20,
"max_depth": 300
},
"logic_flow": [
{
"function": "Multiplier",
"function_params": {
"left_arg": 3.5,
"right_arg": {
"size": 3
}
}
}
]
}

from classiq import ModelDesigner
from classiq.builtin_functions import Multiplier
from classiq.interface.generator.arith.arithmetic import RegisterUserInput

params = Multiplier(left_arg=3.5, right_arg=RegisterUserInput(size=3))
model_designer = ModelDesigner()
model_designer.Multiplier(params)
circuit = model_designer.synthesize()
circuit.show()


This code example generates a circuit that multiplies 2 arguments. Here "left_arg" is a fixed-point number $$(11.1)_2$$ and "right_arg" a quantum register of size 3.

#### Generated Circuit¶ 