# 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

## Examples

#### Example 1: Subtraction of Two Quantum Variables

This example generates a quantum program that subtracts one quantum variables from the other, both of size 3 qubits.

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, "subtraction_2vars_example")
qprog = synthesize(qmod)

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

[{'a': 4.0, 'b': 5.0, 'res': -1.0}: 1000]


#### Example 2: Subtraction of a Float from a Register

This example generates a quantum program which subtracts two argument. The left_arg is defined to be a fix point number $$(11.1)_2$$ (3.5). The right_arg is defined to be a quantum register of size of three.

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

@qfunc
def main(a: Output[QNum], res: Output[QNum]) -> None:
prepare_int(4, a)
res |= a - 3.5

qmod = create_model(main)

from classiq import execute, synthesize, write_qmod

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

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

[{'a': 4.0, 'res': 0.5}: 1000]