Encodings Reference¶

The Quantum Encoding Atlas implements 16 quantum data encoding methods with a unified API across PennyLane, Qiskit, and Cirq backends. Each encoding transforms classical feature vectors into quantum states using different circuit structures, offering distinct tradeoffs in expressibility, trainability, hardware cost, and noise resilience.

At a Glance¶

Encoding	Qubits	Entangling	Depth	Best For
Angle	n	No	O(1)	Low-noise baselines, few features
Amplitude	log n	Yes	O(2^n)	Qubit-limited regimes, many features
Basis	n	No	O(1)	Binary / categorical data
IQP	n	Yes	O(n^2)	Quantum kernels, provable hardness
ZZ Feature Map	n	Yes	O(n^2)	Pairwise interactions, kernel methods
Pauli Feature Map	n	Yes	O(n^2)	Configurable Pauli rotations
Data Re-uploading	n	Yes	O(L)	Universal approximation, small circuits
Hardware Efficient	n	Yes	O(L)	NISQ devices, native gate sets
Higher-Order Angle	n	No	O(n^d)	Polynomial features without entanglement
QAOA-Inspired	n	Yes	O(L)	Combinatorial structure in data
Hamiltonian	n	Yes	O(L)	Physics-informed encoding
Trainable	n	Yes	O(L)	Learnable encoding layers
Symmetry-Inspired	n	Yes	O(L)	Heuristic symmetry-aware bias
SO(2) Equivariant	n	Yes	O(L)	2D rotational symmetry
Cyclic Equivariant	n	Yes	O(L)	Z_n cyclic shift symmetry
Swap Equivariant	n	Yes	O(L)	S_2 pair-swap symmetry

Choosing an Encoding¶

Not sure where to start? The Decision Guide walks you through selecting an encoding based on your data, hardware, and priorities.

As a quick rule of thumb:

                    Do your features have known symmetry?
                    ├── Yes ──► SO(2) / Cyclic / Swap Equivariant
                    └── No
                         │
                    Do you need provable quantum advantage?
                    ├── Yes ──► IQP / ZZ Feature Map
                    └── No
                         │
                    Are you qubit-limited?
                    ├── Yes ──► Amplitude Encoding
                    └── No
                         │
                    Do you want a trainable feature map?
                    ├── Yes ──► Data Re-uploading / Trainable
                    └── No ──► Angle Encoding (safe default)

Taxonomy¶

By Circuit Structure¶

Product-state encodings — no entanglement, classically simulable:

Angle Encoding — single-qubit rotations
Basis Encoding — computational basis mapping
Higher-Order Angle — polynomial rotations

Diagonal-phase encodings — Hadamard + phase + entanglement layers:

IQP Encoding — provably hard to simulate
ZZ Feature Map — pairwise ZZ interactions
Pauli Feature Map — configurable Pauli strings

Variational encodings — interleaved data and parametric layers:

Data Re-uploading — repeated data injection
Hardware Efficient — native gate sets
Trainable Encoding — learnable parameters
QAOA-Inspired — cost-mixer structure

Global state preparation — amplitude-level encoding:

Amplitude Encoding — exponential compression

Equivariant encodings — symmetry-preserving circuits:

SO(2) Equivariant — continuous rotations
Cyclic Equivariant — discrete cyclic shifts
Swap Equivariant — pairwise permutations
Symmetry-Inspired — heuristic symmetry

By Simulability¶

Classically Simulable	Not Classically Simulable
Angle, Basis, Higher-Order Angle	IQP, ZZ, Pauli, Amplitude, Data Re-uploading, Hardware Efficient, QAOA, Hamiltonian, Trainable, all Equivariant maps

Unified API¶

Every encoding shares the same interface:

from encoding_atlas import IQPEncoding

enc = IQPEncoding(n_features=4, reps=2)

# Properties
enc.n_qubits          # Number of qubits
enc.depth             # Circuit depth
enc.properties        # Full EncodingProperties dataclass

# Circuit generation
circuit = enc.get_circuit(x, backend='pennylane')

# Configuration
enc.config            # Read-only dict of parameters

See the Quick Start for a hands-on introduction and the API Reference for the complete programmatic interface.