> ## 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.
# Swap Test Algorithm
Open this notebook in GitHub to run it yourself
The swap test is a quantum function that checks the overlap between two quantum states.
The inputs of the function are two quantum registers of the same size, $|\psi_1\rangle, \,|\psi_2\rangle$, and it returns as output a single test qubit whose state encodes the overlap between the two inputs: $|q\rangle_{\rm test} = \alpha|0\rangle + \sqrt{1-\alpha^2}|1\rangle$, with
$$
\alpha^2 = \frac{1}{2}\left(1+|\langle \psi_1 |\psi_2 \rangle |^2\right).
$$
Thus, the probability of measuring the test qubit at state $|0\rangle$ is $1$ if the states are identical (up to a global phase) and 0.5 if the states are orthogonal to each other.
The quantum model starts with an $H$ gate on the test qubit, followed by swapping between the two states controlled on the test qubit (a controlled-SWAP gate for each of the qubits in the two states) and a final $H$ gate on the test qubit.
A general scheme of the swap test algorithm:
Prepare two random states:
```python theme={null}
import numpy as np
np.random.seed(12)
NUM_QUBITS = 3
amps1 = 1 - 2 * np.random.rand(
2**NUM_QUBITS
) # vector of 2^3 numbers in the range [-1,1]
amps2 = 1 - 2 * np.random.rand(2**NUM_QUBITS)
amps1 = amps1 / np.linalg.norm(amps1) # normalize the vector
amps2 = amps2 / np.linalg.norm(amps2)
```
Create a model and synthesize:
```python theme={null}
from classiq import *
@qfunc
def main(test: Output[QBit]):
state1 = QArray("state1")
state2 = QArray("state2")
prepare_amplitudes(amps1.tolist(), 0.0, state1)
prepare_amplitudes(amps2.tolist(), 0.0, state2)
swap_test(state1, state2, test)
drop(state1)
drop(state2)
qmod = create_model(main)
qmod = set_execution_preferences(qmod, num_shots=100_000)
qprog = synthesize(qmod)
show(qprog)
```
**Output:**
```
Quantum program link: https://platform.classiq.io/circuit/39FfXKGSkewL94nnWJoZyyQt6Ks
```
**Output:**
```
https://platform.classiq.io/circuit/39FfXKGSkewL94nnWJoZyyQt6Ks?login=True&version=17
```
## Swap Test Qmod Implementations
The swap test is defined as a library function in the Qmod language.
Verify the results:
```python theme={null}
result = execute(qprog).result_value()
```
## Comparing Measured with Exact Overlap
Using the expected probability of measuring the state $|0\rangle$ as defined above,
$$
\alpha^2 = \frac{1}{2}\left(1+|\langle \psi_1 |\psi_2 \rangle |^2\right),
$$
we extract the overlap $|\langle \psi_1 |\psi_2 \rangle |=\sqrt{2 P\left(q_{\text{test}}=|0\rangle\right)-1}$.
The exact overlap is computed with the dot product of the two state vectors.
Note that for the sake of this demonstration we execute this circuit $100,000$ times to improve the precision of the probability estimate.
This is usually not required in actual programs.
```python theme={null}
overlap_from_swap_test = np.sqrt(
2 * result.counts["0"] / sum(result.counts.values()) - 1
)
exact_overlap = np.abs(amps1 @ amps2)
```
```python theme={null}
print("States overlap from Swap-Test result:", overlap_from_swap_test)
print("States overlap from classical calculation:", exact_overlap)
```
**Output:**
```
States overlap from Swap-Test result: 0.4762352359916262
States overlap from classical calculation: 0.46972037234759095
```
```python theme={null}
RTOL = 0.05
assert np.isclose(
overlap_from_swap_test, exact_overlap, RTOL
), f"""
The quantum result is too far from the classical one by a relative tolerance of {RTOL}.
Please verify your parameters"""
```