Geometric Differences between Classical and Quantum Kernels

Classical kernel methods, like Support Vector Machines (SVMs), implicitly map input data $x$ into a higher-dimensional feature space $\mathcal{F}_{classical}$ via a mapping $\phi_{classical}(x)$. The kernel function $k_{classical}(x, x')$ then efficiently calculates the inner product $\langle \phi_{classical}(x), \phi_{classical}(x') \rangle$ in this space, without needing to explicitly compute the coordinates of $\phi_{classical}(x)$. The geometry of this classical feature space $\mathcal{F}_{classical}$ is directly determined by the choice of the kernel function. For instance:

• Linear Kernel: $k(x, x') = x^T x'$. The feature space is the same as the input space. Geometry is Euclidean.
• Polynomial Kernel: $k(x, x') = (\gamma x^T x' + r)^d$. The feature space consists of monomials of the input features up to degree $d$. The mapping creates curved decision boundaries in the original space by finding linear boundaries in this higher-dimensional polynomial feature space.
• Radial Basis Function (RBF) Kernel: $k(x, x') = \exp(-\gamma \|x - x'\|^2)$. This corresponds to an infinite-dimensional feature space. Geometrically, it measures similarity based on Euclidean distance, creating localized "bumps" of influence around data points.

Quantum kernels operate on a similar principle but utilize a quantum feature map $|\phi(x)\rangle$ to embed data into a quantum feature space, the Hilbert space $\mathcal{H}$ of a multi-qubit system. The quantum kernel is typically computed from the inner product of these quantum states, often as $k_{quantum}(x, x') = |\langle \phi(x) | \phi(x') \rangle|^2$. This difference in the underlying space and the method of calculating similarity leads to significant geometric distinctions compared to classical kernels.

Dimensionality and Structure

The most immediate difference lies in the potential dimensionality. For an $N$-qubit system, the Hilbert space $\mathcal{H}$ has dimension $2^N$. This exponential scaling offers a potentially vast space for representing data.

Comparison of classical and quantum kernel mappings. Quantum feature maps embed data into an exponentially large Hilbert space, calculating kernels via quantum state overlap.

While classical kernels like RBF also map to infinite dimensions, the structure imposed by quantum mechanics is fundamentally different. The geometry of $\mathcal{H}$ is not just "large"; it's structured by the principles of superposition and entanglement generated by the quantum feature map circuit $U_{\phi(x)}$.

• Superposition: Allows feature vectors $|\phi(x)\rangle$ to exist in combinations of basis states simultaneously.
• Entanglement: Creates correlations between qubits (features in $\mathcal{H}$) that have no classical analogue. This can lead to complex geometric relationships between the embedded data points $|\phi(x)\rangle$.

The specific geometry depends entirely on the choice of the quantum circuit $U_{\phi(x)}$ used for encoding. A simple product state encoding might create a relatively simple geometry, whereas a circuit with significant entangling capabilities can create intricate structures within the Hilbert space.

Inner Products and Similarity

Classical kernels directly compute the inner product $\langle \phi_{classical}(x), \phi_{classical}(x') \rangle$. This value relates directly to the angle between the feature vectors in $\mathcal{F}_{classical}$ and often serves as a direct measure of similarity.

Quantum kernels typically use the squared magnitude of the inner product, $k(x, x') = |\langle \phi(x) | \phi(x') \rangle|^2$. This quantity is related to the probability of transitioning from state $|\phi(x)\rangle$ to $|\phi(x')\rangle$ (or measuring one state in the basis defined by the other). This squaring introduces an additional non-linearity compared to the direct inner product. Geometrically, it means the quantum kernel is sensitive not just to the angle between state vectors in Hilbert space but to the overlap probability. Two pairs of states could have the same angle between them but different kernel values if their global phase differs, although this phase is often irrelevant when using the squared overlap. More importantly, the mapping $x \rightarrow |\phi(x)\rangle$ itself is highly non-linear due to the nature of quantum gates.

Separability and Expressivity

The hope is that the unique geometry of the quantum feature space allows for better separation of complex datasets. By mapping data into this high-dimensional, structured space, linear separation (as performed by SVM) in $\mathcal{H}$ might correspond to highly non-linear separation in the original input space.

• Classical: The separating power is determined by the chosen kernel function (linear, polynomial, RBF). The geometry is fixed by this choice.
• Quantum: The separating power depends on the expressivity of the quantum feature map circuit $U_{\phi(x)}$. More complex circuits with greater entangling power can, in principle, generate more complex geometries and potentially separate more intricate patterns. However, this capability is not guaranteed and comes with challenges. Overly complex or deep circuits can lead to the kernel concentration phenomenon (discussed in the next section), where all kernel values concentrate around a specific value, making discrimination impossible.

The geometric differences imply that quantum kernels might excel at tasks where the underlying data structure aligns well with the geometry induced by quantum evolution (superposition and entanglement), potentially capturing correlations missed by classical kernels. Conversely, classical kernels might be more effective or efficient for problems whose structure naturally fits polynomial or Gaussian relationships. Understanding and designing feature maps $U_{\phi(x)}$ that induce problem-specific, advantageous geometries is a primary focus in quantum kernel research.