Quantum Feature Maps

Quantum Feature Maps

A quantum feature map is a parameterized quantum circuit that encodes classical data x into a quantum state $|\varphi (x)⟩$. This is the foundation of quantum kernel methods and quantum neural networks.

Why Feature Maps Matter

In classical machine learning, kernel methods avoid the curse of dimensionality by implicitly computing inner products in a high-dimensional feature space. The power comes from the feature map $\varphi (x)$: the higher-dimensional space, the richer the structure you can capture.

Quantum feature maps take this to an extreme: an n-qubit quantum computer operates in a $2^n$-dimensional Hilbert space. Even modest qubit counts give feature spaces that are impossible to represent explicitly on classical hardware.

The ZZ Feature Map (Pauli Feature Map)

The most commonly studied quantum feature map for kernel experiments is the ZZ feature map (also called the second-order Pauli feature map):

Layer 1 (Hadamard): Apply H to all qubits Layer 2 (Single-qubit): For each qubit i, apply $R_z(2x_i)$ Layer 3 (Two-qubit): For each pair (i, j): cx(i,j), $R_z(2{\varphi }_{ij}(x))$, cx(i,j)

where ${\varphi }_{ij}(x)=(\pi -x_i)(\pi -x_j)$.

Repeat layers 1-3 twice (reps=2) for a deeper feature map.

The pairwise interaction terms ${\varphi }_{ij}(x)=(\pi -x_i)(\pi -x_j)$ encode correlations between features. These cross terms are genuinely nonlinear and potentially hard for classical RBF kernels to capture.

Geometric Interpretation

Each data point x maps to a point on the surface of the $2^n$-dimensional Hilbert space (a complex unit sphere). Two data points are "similar" if their quantum states have high overlap.

The geometry of the quantum feature space, which points are close and which are far, is determined by the feature map circuit. Designing feature maps with good geometry for a specific task is an open research problem.

Hardware-Efficient Feature Maps

For near-term hardware, feature maps must be shallow. A hardware-efficient feature map uses only native gate operations: