Pre Lab

Learning Objectives

• Explain the main idea of Support Vector Machines (SVM).
  
  

• Describe the margin, support vectors, and the role of the kernel function.
  
  

• Explain how Quantum SVM (QSVM) replaces the classical kernel with a quantum kernel.
  
  

• Understand that both SVM and QSVM use the same optimization idea but different ways to measure similarity between data points.

1. What is SVM?

Support Vector Machine (SVM) is a classical machine learning model for classification.

Each data point is written as a feature vector

• X ∈ ℝᵈ

with a label

• y ∈ {0,1}

SVM tries to find a decision function 

such that the two classes are separated as well as possible. The sign of f(x) gives the prediction: 

if f (x) ≥ 0, predict class +1

if f(x) < 0, predict class -1

The key idea is to find a hyperplane that maximizes the margin, which means the distance between the hyperplane and the closest points from each class. In the hard-margin case, this can be written as the optimization problem

2. The Kernel Trick

Many real-world datasets are not linearly separable in the original feature space. To handle this, SVM uses a kernel function

where ϕ(x) maps the data into a high-dimensional feature space. 

Common choices include:

• Linear kernel

• Polynomial kernel

• Radial Basis Function (RBF) kernel

The important point is that we never need to compute ϕ(x) directly—SVM only needs the kernel values K(xi​,xj​). This is called the kernel trick.

3. What is QSVM? 

Quantum Support Vector Machine (QSVM) keeps the same SVM optimization framework, but uses a quantum kernel instead of a classical kernel.

We first encode each classical data point x into a quantum state  |ψ(x)⟩.

Then we define the quantum kernel as a similarity between two quantum states

This kernel is estimated by running a quantum circuit on a simulator or quantum device.

Once we have the full kernel matrix built from above, we plug it into the standard SVM solver, just like a classical kernel.