Data Encoding Circuits (Feature Maps)

qc/ml

A feature map (or encoding circuit) is what turns a classical data point $x$ into a quantum state $|\Psi (x)⟩$ before anything “quantum” can happen to it — every quantum machine learning technique needs one. Mechanically it’s a parameterized circuit exactly like the ones VQE/QAOA use, except the bound values aren’t free optimization variables — they’re (transformations of) the classical input itself. The simplest version is angle encoding: feed $x$ directly into a rotation gate, e.g. $R_y(x)|0⟩$. Key insight: naive angle encoding wastes the circuit’s expressive range, since a single rotation only covers one period; Chebyshev encoding (used by sQUlearn’s ChebyshevPQC) instead scales the input through Chebyshev polynomials before it hits the rotation angles, spreading the data non-linearly across multiple qubits/layers — a form of data re-uploading, where the same classical value gets fed back into the circuit at several points rather than just once, which measurably increases the model’s expressive capacity for the same qubit count.

from qiskit.circuit import Parameter
import numpy as np
 
x = Parameter('x')
# naive angle encoding: one rotation, one period of coverage
qc.ry(x, 0)
 
# Chebyshev-style: re-scale through a polynomial before the rotation,
# then re-upload x again in a later layer — same value, more expressive coverage
qc.ry(2 * np.arccos(x), 0)   # T_1(x)-style rescaling, illustrative

Related

• Parameterized Circuits — same “structure once, bind values” substrate; here the bound values come from data, not free optimization parameters
• Quantum Neural Networks (QNN) — the model built on top of this encoding
• Quantum Kernel Methods — the other model built on top of this encoding

Self-Check

• How is a feature map’s Parameter binding different from how VQE binds its ansatz parameters, given both use the exact same Parameter/ParameterVector machinery?
• Why does naive single-rotation angle encoding limit a circuit’s expressive range, and how does Chebyshev/data-re-uploading encoding address that?
• Could you explain “data re-uploading” to someone who only knows angle encoding as “feed x into a rotation gate”?