Variational

QAOA Encoding

QAOA-inspired encoding with alternating cost and mixer layers for combinatorial data.

Qubits

Depth

Total Gates

Simulability

Not simulable

Mathematical Formulation

|\psi(\mathbf{x})\rangle = \prod_{p=1}^{\text{reps}} U_M(\beta) \cdot U_C(\gamma, \mathbf{x}) |+\rangle^{\otimes n}

Description

QAOA (Quantum Approximate Optimization Algorithm) encoding adapts the QAOA circuit structure for data encoding. It alternates between cost layers that encode features through data-dependent rotations and mixer layers that spread information across qubits via entangling gates. This structure is particularly well-suited for combinatorial optimization problems.

The circuit begins with Hadamard gates to create equal superposition, then repeats p layers of: (1) data rotation gates R_data(γ·x_i) encoding features as phase-like rotations, (2) entangling gates (CZ, CX, or RZZ) coupling qubits, and (3) mixer rotation gates R_mixer(β) for exploration. The γ and β parameters control the encoding strength and mixing, respectively.

The encoding supports both linear and quadratic feature maps: linear maps φ(x_i) = γ·x_i, while quadratic uses φ(x_i) = γ·x_i². An edge-coloring-based algorithm optimizes the circuit depth for the entanglement pattern, yielding near-optimal parallel gate scheduling.

Circuit Diagram

Property Radar

Properties

Qubits

Circuit Depth

Total Gates

Single-Qubit Gates

Two-Qubit Gates

Parameters

Entangling

Yes

Simulability

Not Simulable

Expressibility

—

Entanglement Capability

—

Trainability

0.77

Noise Resilience

—

Resource Scaling

How resource requirements grow with the number of input features.

Features	Qubits	Depth	Gates	2Q Gates
2	2	7	12	2
4	4	9	26	6
8	8	9	54	14
16	16	9	110	30

Code Examples

QAOA encoding with PennyLane using linear entanglement and CZ gates.

python

from encoding_atlas import QAOAEncoding
import pennylane as qml
import numpy as np

enc = QAOAEncoding(n_features=4, reps=2, entanglement="linear")
dev = qml.device("default.qubit", wires=enc.n_qubits)

@qml.qnode(dev)
def circuit(x):
    enc.get_circuit(x, backend="pennylane")
    return qml.state()

x = np.array([0.1, 0.5, 1.2, 2.3])
state = circuit(x)

When to Use This Encoding

Encoding data for combinatorial optimization problems
Graph-structured data encoding (e.g., MaxCut, TSP)
QAOA-inspired variational classifiers
Problems with natural cost-function structure
Quantum optimization benchmarks

Pros & Cons

Advantages

Natural fit for combinatorial optimization problems
Flexible entangling gate choices (CX, CZ, RZZ)
Edge-coloring optimized depth for parallel gate scheduling
Supports both linear and quadratic feature maps
Good trainability with linear entanglement

Limitations

Specialized structure — may not be optimal for general ML tasks
Full entanglement scales poorly for large feature counts
Fixed γ and β parameters limit adaptability without optimization
Barren plateau risk increases with reps > 10
Not designed for continuous feature-intensive tasks

References

[1]Farhi, E., Goldstone, J., & Gutmann, S. (2014). A Quantum Approximate Optimization Algorithm. arXiv:1411.4028.
[2]Hadfield, S., et al. (2019). From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz. Algorithms, 12(2), 34.
[3]Zhou, L., et al. (2020). Quantum Approximate Optimization Algorithm: Performance, Mechanism, and Implementation on Near-Term Devices. Physical Review X, 10(2), 021067.