# Modulo

The modulo operation (denoted as '%') returns the remainder (called "modulus") of a division. Given two numbers $$a$$ and $$n$$, the result of ($$a \% n$$) is the remainder of the division of a by n. The modulo operation is supported only for $$n = 2^m$$ for an integer $$m$$, its result is the $$m$$ least significant bits.

For example, the binary representation of the number $$53$$ is $$0b110101$$. The expression ($$53 \% 8$$) equals $$0b101 = 5$$, because $$8 = 2^3$$, which means only accounting for the $$3$$ least significant bits of $$53$$.

#### Implementation in Expressions

If an expression is defined using a modulo operation, the output size is set recursively to all of its subexpressions. But if for some sub-expressions, another modulo operation is used, the sub-expression's output_size is determined by the minimal value between the output_size of the sub-expression and the expression.

See this example: $$(((a + b) \% 4) + (c + d)) \% 8$$. The result of expression $$a + b$$ is saved on a two-qubit register, and the results of expressions $$c + d$$ and $$((a + b) \% 4) + (c + d)$$ are saved using three qubits each.

# Example

This example generates a quantum program that adds two five-qubit arguments: a on qubits 0-4, and b on qubits 5-9. The adder result should have been calculated on a 6-qubit register. However, the modulo operation decides that the output register of the adder only contains its two least significant qubits. Thus, the adder result is written to a two-qubit register, on qubits 10-11.

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

@qfunc
def main(a: Output[QNum], b: Output[QNum], res: Output[QNum]) -> None:
allocate(5, a)
allocate(5, b)

inplace_prepare_int(4, a)
inplace_prepare_int(7, b)

res |= (a + b) % 4

qmod = create_model(main)

from classiq import execute, synthesize, write_qmod

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

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

[{'a': 4.0, 'b': 7.0, 'res': 3.0}: 1000]