Projected Quantum Kernel

Overview

Projects the quantum kernel to a lower-dimensional classical space for improved computational efficiency and interpretability. Instead of full fidelity, uses expectation values of observables.

Property	Value
Category	Machine Learning
Difficulty	Advanced
Framework	PennyLane
Qubits	4
Depth	~8
Gates	H, RZ, CNOT

The Method

Standard Quantum Kernel

CODE
K(x, y) = |⟨φ(x)|φ(y)⟩|²

Projected Kernel

CODE
K(x, y) = ⟨p(x), p(y)⟩

Where p(x) = [⟨O₁⟩ₓ, ⟨O₂⟩ₓ, ..., ⟨Oₘ⟩ₓ] is a vector of expectation values.

IQP Feature Map

CODE
┌───┐┌────────┐     ┌─────────────┐┌───┐
q_0: ┤ H ├┤ RZ(x₀) ├──■──┤ RZ(x₀x₁)   ├┤ H ├
     ├───┤├────────┤┌─┴─┐└─────────────┘├───┤
q_1: ┤ H ├┤ RZ(x₁) ├┤ X ├───────────────┤ H ├
     ├───┤├────────┤└───┘               ├───┤
...

Running the Circuit

PYTHON
from circuit import run_circuit

# Random projections
result = run_circuit(n_projections=8, projection_type='random')

# Basis projections
result = run_circuit(n_projections=8, projection_type='basis')

print(f"Projection dimension: {result['projection_dim']}")
print(f"Kernel rank: {result['kernel_rank']}")

Projection Types

Random Pauli

Measures random Pauli strings
Captures diverse correlations
O(m) measurements needed

Basis Projections

Measures computational basis probabilities
More interpretable
Limited to 2ⁿ projections

Advantages

Aspect	Full Kernel	Projected Kernel
Measurements	O(1)	O(m)
Classical dimension	1	m
Interpretability	Low	Higher
Error sensitivity	High	Lower

Applications

Large-scale ML: More efficient than full kernel
Noisy devices: More robust to errors
Feature analysis: Interpretable components