Quantum Kernels

Quantum Kernels

Kernel Methods in Classical ML

A kernel function K(x, x') measures similarity between two data points. In classical ML, kernel methods, especially Support Vector Machines (SVMs), use kernels to classify data by finding optimal decision boundaries in a (possibly infinite-dimensional) feature space.

The kernel trick: instead of explicitly computing the feature map $\varphi (x)$, compute the dot product $K(x,x^{\prime })=\varphi (x)\cdot \varphi (x^{\prime })$ directly. This allows working in very high-dimensional spaces without the computational cost of explicit feature computation.

The choice of kernel encodes assumptions about data structure. The RBF kernel assumes nearby points are similar with exponentially decaying similarity. Polynomial kernels capture interaction effects.

Quantum Feature Maps

A quantum feature map $\varphi (x)$ encodes classical data into a quantum state:

where $U(x)$ is a parameterized quantum circuit that depends on the input data x. Each data point maps to a quantum state in an exponentially large Hilbert space ($2^n$ dimensions for n qubits).

The quantum kernel is the inner product between two such states:

Measuring this quantity requires running a circuit and checking the probability of the all-zeros outcome. This probability equals the kernel value.

The ZZ Feature Map as a Kernel

You already built the standard construction in the Quantum Feature Maps lesson earlier in this track: H on every qubit, single-qubit $R_z(2x_i)$ rotations, then pairwise cx-$R_z(2{\varphi }_{ij}(x))$-cx blocks with ${\varphi }_{ij}(x)=(\pi -x_i)(\pi -x_j)$, repeated twice. Plugged into the definition above, that circuit is $U(x)$: its pairwise cross-terms are what make the resulting kernel hard to reproduce classically.