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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.


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 (

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