# Quantum Embedding Kernels

## Brief Explanation

This research paper introduces the concept of Quantum Embedding Kernels (QEKs), which are a type of quantum kernel method for machine learning on near-term quantum computers. QEKs use parameterized quantum circuits to embed data into the Hilbert space of quantum states, and then compute the inner product between these embedded states, which serves as the kernel function.

The paper discusses the theory behind QEKs, how to optimize their variational parameters using the kernel-target alignment, and techniques to mitigate the effects of noise from quantum devices.
## Example

### Quantum Feature Map

A quantum feature map is a quantum circuit that embeds a data point `x` into a quantum state `|φ(x)⟩`. For example, a simple quantum feature map could be:

def quantum_feature_map(x, wires):
    qml.RX(x[0], wires=wires[0])
    qml.RY(x[1], wires=wires[1])

This circuit takes a 2-dimensional data point `x` and encodes it into a quantum state using single-qubit rotations.

## Quantum Embedding Kernel

The QEK associated with the quantum feature map `|φ(x)⟩` is defined as:

k(x, x0) = |⟨φ(x0)|φ(x)⟩|²

This kernel function computes the overlap between the quantum states corresponding to the datapoints `x` and `x0`

## Problem Statement
Develop a quantum machine learning model using Quantum Embedding Kernels (QEKs) for a binary classification task on any dataset of your choice.

Expected Outcome:

- Implementation of a parameterized quantum circuit for embedding data into quantum states (quantum feature map).
- Computation of the QEK matrix for the given dataset.
- Optimization of the variational parameters of the quantum feature map by maximizing the kernel-target alignment.
- Application of noise mitigation and regularization techniques to improve the quality of the QEK matrix.
- Training of a support vector machine (SVM) classifier using the optimized QEK matrix.
- Evaluation of the classification accuracy of the SVM classifier on a test set.

## Resources

https://arxiv.org/pdf/2105.02276
https://pennylane.ai/qml/demos/tutorial_kernels_module/