Quantum Kernel Estimation & High-Dimensional Feature Maps

Quantum Kernel Estimation (QKE) projects classical feature vectors into a high-dimensional quantum Hilbert space $\mathcal{F}$ using a quantum feature map $\Phi(x)$. The inner product between two quantum states serves as a kernel matrix for Support Vector Machines (SVMs):

$$K(x_i, x_j) = |\langle \Phi(x_i) | \Phi(x_j) \rangle|^2$$

Quantum Advantage in Kernel Methods

When $\Phi(x)$ involves quantum circuits that are hard to compute classically (such as IQP or Hamiltonian evolution feature maps), the quantum kernel provides computational leverage over classical RBF or polynomial kernels.

# Quantum Kernel Evaluation
@qml.qnode(dev)
def kernel_circuit(x1, x2):
    feature_map(x1)
    qml.adjoint(feature_map)(x2)
    return qml.probs(wires=range(wires))