> ## Documentation Index
> Fetch the complete documentation index at: https://prod-mint.classiq.io/llms.txt
> Use this file to discover all available pages before exploring further.

# Modular Exponentiation

<Card title="View on GitHub" icon="github" href="https://github.com/Classiq/classiq-library/blob/main/functions/function_usage_examples/arithmetic/modular_exp/modular_exp_example.ipynb">
  Open this notebook in GitHub to run it yourself
</Card>

The `modular_exp` function raises a classical integer `a` to the power of a quantum number `power` modulo classical integer `n`, times a quantum number `x`.

The function performs:

$$
|x\rangle |power\rangle = |(a^{power} \mod n)\cdot x\rangle | power\rangle
$$

Specifically if at the input $x=1$, at the output $x=a^{power} \mod n$.

## Example

This example generates a quantum program that initializes a `power` variable with a uniform superposition, and exponentiate the classical value `A` with `power` as the exponent, in superposition.

The result is calculated inplace to the variable `x` module `N`.

Notice that `x` should have size of at least $\lceil(log_2(N) \rceil$, so it is first allocated with a fixed size, then initialized with the value '1'.

```python theme={null}
import numpy as np

from classiq import *
from classiq.qmod.symbolic import ceiling, log

# constants
N = 5
A = 4


@qfunc
def main(power: Output[QNum], x: Output[QNum]) -> None:
    allocate(ceiling(log(N, 2)), x)
    x ^= 1

    # initialize a uniform superposition of powers
    allocate(3, power)
    hadamard_transform(power)

    modular_exp(n=N, a=A, x=x, power=power)


qmod = create_model(main)
qmod = create_model(main)
qprog = synthesize(qmod)
```

```python theme={null}

result = execute(qprog).result_value()
result.parsed_counts
```

<Info>
  **Output:**

  ```
  [{'power': 3, 'x': 4}: 136,
     {'power': 1, 'x': 4}: 135,
     {'power': 7, 'x': 4}: 130,
     {'power': 5, 'x': 4}: 124,
     {'power': 2, 'x': 1}: 124,
     {'power': 4, 'x': 1}: 123,
     {'power': 6, 'x': 1}: 117,
     {'power': 0, 'x': 1}: 111]
    

  ```
</Info>

Verify all results are as expected:

```python theme={null}
assert np.all(
    [
        count.state["x"] == (A ** count.state["power"] % N)
        for count in result.parsed_counts
    ]
)
```