Quantum Kernels in Depth

A quantum kernel measures similarity between data points by embedding them as quantum states and taking their overlap. The classical learner — typically a support-vector machine — never sees the quantum state; it only sees the kernel matrix. The whole quantum content lives in the feature map.

Feature map and kernel

A feature map is a fixed (untrained) circuit that turns a classical vector $\boldsymbol{x}$ into a state:

The kernel is the squared overlap of two such states,

which is a positive-definite similarity measure — exactly what a kernel SVM requires. Havlíček et al. (2019) and Schuld & Killoran (2018) established this feature-space view of quantum models.

Angle encoding

The simplest useful feature map is angle encoding: rotate each qubit by an angle proportional to a feature. With one qubit per feature and a $Y$-rotation,

The factor of $2$ is conventional: with $R_Y(2x_i)$ the qubit's amplitude on $|1\rangle$ becomes $\sin(x_i)$, so the feature $x_i$ controls the rotation angle directly. For a single qubit,

For a two-feature point $\boldsymbol{x}=(x_0,x_1)$ on two qubits the encoded state is the tensor product

Computing a kernel entry on hardware

You never have the amplitudes directly. The standard trick is the overlap (inversion) test: run $U_\Phi(\boldsymbol{x}')^\dagger U_\Phi(\boldsymbol{x})$ on $|0\rangle$ and measure the probability of returning to $|0\rangle^{\otimes n}$ — that probability is $k(\boldsymbol{x},\boldsymbol{x}')$. The kernel matrix is filled in by repeating this over all pairs, then handed to a classical SVM.

Try it

Build the angle-encoding feature map state for the two-feature data point $\boldsymbol{x} = (0.6,\ 1.2)$ on two qubits, using $R_Y(2x_i)$ on qubit $i$. The grader checks the full state vector, so the amplitudes must match $\cos(0.6)\cos(1.2),\ \cos(0.6)\sin(1.2),\ \sin(0.6)\cos(1.2),\ \sin(0.6)\sin(1.2)$.

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