# Multiplication

The multiplication operation, denoted '$$*$$', is a series of additions ("long multiplication"). The multiplier has different implementations, depending on the type of adder in use.

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

## Examples

The calculation of -5 * 3 = -15.

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

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

## Examples

#### Example 1: Two Quantum Variables Multiplication

This code example generates a quantum program that multiplies two arguments. Both of them are defined as quantum variables of size 3.

from classiq import Output, QArray, QBit, QNum, create_model, prepare_int, qfunc

@qfunc
def main(a: Output[QNum], b: Output[QNum], res: Output[QNum]) -> None:
prepare_int(4, a)
prepare_int(5, b)
res |= a * b

qmod = create_model(main)

from classiq import execute, synthesize, write_qmod

write_qmod(qmod, "multiplication_2vars_example")
qprog = synthesize(qmod)

result = execute(qprog).result()[0].value
result.parsed_counts

[{'a': 4.0, 'b': 5.0, 'res': 20.0}: 1000]


#### Example 2: Float and Quantum Variable Multiplication

This code example generates a quantum program that multiplies two arguments. Here, the left argument is a fixed-point number $$(11.1)_2$$ (3.5), and the right argument is a quantum variable of size 2.

from classiq import (
Output,
QArray,
QBit,
QNum,
allocate,
create_model,
qfunc,
)

@qfunc
def main(a: Output[QNum], res: Output[QNum]) -> None:
allocate(2, a)

res |= 3.5 * a

qmod = create_model(main)

from classiq import execute, synthesize, write_qmod

write_qmod(qmod, "multiplication_float_example")
qprog = synthesize(qmod)

result = execute(qprog).result()[0].value
result.parsed_counts

[{'a': 0.0, 'res': 0.0}: 270,
{'a': 3.0, 'res': 10.5}: 251,
{'a': 2.0, 'res': 7.0}: 241,
{'a': 1.0, 'res': 3.5}: 238]