> ## 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.

# Exponential State Preparation

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

The `prepare_exponential_state` function
creates a state with exponentially decreasing amplitudes. Namely,
the probability for a state representing an integer $n$ is

$$
P\left(n\right) = \frac{1}{Z} e^{-\lambda n}
$$

where $\lambda$ is the rate, and $Z$ is a normalization factor.
If $q$ in the number of qubits, then

$$
Z = \sum_{n=0} ^{n = 2^q - 1} e^{-\lambda n} = \frac{1 - e^{-\lambda 2^q}}{1 - e^{-\lambda}}
$$

Function: `prepare_exponential_state`

Arguments:

* `rate: CReal`
* `q: QArray[QBit]`

Notice that the function acts inplace on the qubits.

## Example

Prepare a state with probabilities:

$$
P\left(n\right) = \frac{1}{Z} e^{-0.1 n}
$$

where  $n$ is in the range $[0, 31]$.

```python theme={null}
from classiq import *


@qfunc
def main(x: Output[QArray[QBit]]):
    allocate(5, x)
    prepare_exponential_state(0.1, x)


qmod = create_model(main)
```

```python theme={null}

qprog = synthesize(qmod)
```