Quantum Machine Learning Kernels

Quantum Kernel Methods

Quantum kernel methods encode classical data into quantum states and use quantum inner products as a kernel function for machine learning classifiers. They are one of the strongest candidates for near-term quantum advantage in ML.

Classical Kernel Methods

A kernel function K(x, y) measures similarity between two data points. Support Vector Machines (SVMs) find a maximum-margin hyperplane in the kernel-induced feature space. The kernel trick allows this without ever computing the high-dimensional feature vectors explicitly.

Classical kernels: RBF (Gaussian), polynomial, sigmoid. These work well for many problems.

Why quantum? A quantum feature map encodes x into a $2^n$ dimensional Hilbert space: exponentially larger than any practical classical feature space. If this encoding captures structure that classical kernels miss, quantum SVMs could outperform classical ones.

Quantum Feature Maps

A quantum feature map φ(x) encodes classical data x into a quantum state: |φ(x)⟩ = U_φ(x)|0⟩

The interaction term x₁·x₂ captures correlations between data dimensions that no classical linear kernel can represent.

Quantum Kernel Matrix

The quantum kernel between two data points is: K(x, y) = |⟨φ(x)|φ(y)⟩|²

On hardware, this is estimated by running two circuits, one to prepare |φ(x)⟩ and one with the inverse of |φ(y)⟩, and measuring P(|0⟩). Shot counts: typically 1000–8192 per kernel entry.

For a dataset of n samples: n² kernel evaluations → n² × 1000 = millions of shots for modest datasets. This is the main practical bottleneck.

From Kernel to Classifier

Once the n×n kernel matrix K is computed, feed it to a classical SVM. The SVM finds support vectors αᵢ ≥ 0 and a bias b such that the decision boundary is: