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PROJECT

# Find the Inner Product of Two Quantum States Using the SWAP Test

In this project, we'll implement the SWAP test to find the inner product of two quantum states. We can use the SWAP test to determine the similarity between the two quantum states encoded on a quantum circuit.

You will learn to:

Create and simulate a quantum circuit using Qiskit.

Visualize the results of quantum simulation using dedicated packages.

Implement a quantum algorithm and verify its operation.

Compare the simulation results of a quantum algorithm with its theoretical calculations.

Skills

Quantum Computing

Data Visualization

Quantum Algorithms

Prerequisites

Basic understanding of Python

Basic knowledge of linear algebra

Basic understanding of quantum computing

Technologies

NumPy

Qiskit

Matplotlib

Project Description

As the technology employed in quantum computing is fundamentally different from classical computing technology, it requires an altogether different class of algorithms for implementing computational tasks. In fact, quantum algorithms is one of the most active areas of research in quantum computing.

The **SWAP test** is a quantum algorithm used to find the similarity between two quantum states encoded on a quantum circuit. Apart from the quantum state registers, the SWAP test has a single qubit which, after the implementation of certain quantum operations, is measured to determine the results. The results of performing the SWAP test on the quantum states $\vert a\rangle$ and $\vert b\rangle$ are described by the following two equations:

$P\vert0\rangle = \frac{1}{2} + \frac{1}{2}\langle a\vert b\rangle ^2$

$P\vert1\rangle = \frac{1}{2} - \frac{1}{2}\langle a\vert b\rangle ^2$

Here, $P\vert0\rangle$ and $P\vert1\rangle$ are the probabilities of getting the states $\vert0\rangle$ and $\vert1\rangle$ respectively after measurement.

Therefore, by using the SWAP test, we can determine the inner product and the similarity of two quantum states by using the following formula:

$\langle a\vert b\rangle = \sqrt{2 P\vert0\rangle - 1}$

Project Tasks

1

Getting Started

Task 1: Import the Required Libraries

Task 2: Create a Quantum Circuit

Task 3: Apply a Hadamard Gate

Task 4: Apply a CSWAP Gate

Task 5: Apply a Measurement Gate

Task 6: Simulate the Quantum Circuit

2

Implement the SWAP Test

Task 7: Create the Data Registers

Task 8: Create the Quantum Circuit

Task 9: Apply the Quantum Gates

Task 10: Simulate the Circuit

3

Analyze the SWAP Test

Task 11: Create a Module swap_test()

Task 12: Test the swap_test() Function

Task 13: Compare the Results with Theoretical Calculations

Congratulations!