HigherOrderAngleEncoding: Complete Feature Demonstration¶
Overview¶
This notebook is the definitive guide to HigherOrderAngleEncoding from the encoding-atlas library. It demonstrates every constructor parameter, property, method, backend, edge case, and integration point.
What is Higher-Order Angle Encoding?
Standard angle encoding maps each feature $x_i$ to a rotation $R_Y(x_i)$ on qubit $i$. Higher-Order Angle Encoding generalises this by computing polynomial interaction terms — products like $x_i \cdot x_j$ (second-order) or $x_i \cdot x_j \cdot x_k$ (third-order) — and distributing them across qubits.
The rotation angle for each qubit is the sum of all terms assigned to it (scaled by a configurable factor):
$$\theta_q = s \cdot \sum_{T \in \mathcal{T}_q} \prod_{i \in T} x_i$$
where $s$ is the scaling factor and $\mathcal{T}_q$ is the set of polynomial terms assigned to qubit $q$ via round-robin distribution.
Key Properties:
Creates feature interactions without entangling gates → product state
Classically simulable (useful when combined with entangling ansatz layers)
Supports three rotation axes (X, Y, Z), two combination methods (product, sum), configurable order, scaling, and repetitions
Table of Contents¶
Installation & Setup
Constructor Parameters & Validation
Core Properties
Polynomial Term Generation & Assignment
Rotation Angle Computation
Circuit Generation — PennyLane Backend
Circuit Generation — Qiskit Backend
Circuit Generation — Cirq Backend
Batch Circuit Generation & Parallel Processing
Gate Count Breakdown
Resource Summary
EncodingProperties (Base Class)
Capability Protocols
Registry & Factory Access
Encoding Guide & Recommendations
Equality, Hashing & Collections
String Representation
Serialization (Pickle)
Thread Safety & Concurrent Access
Edge Cases & Numerical Stability
Comparison: product vs sum Combination
Comparison: With vs Without First-Order Terms
Scaling Behaviour
Term Count Utility Function
Logging & Debugging
Visualization: Comparing Encodings
Analysis Tools Integration
Summary
1. Installation & Setup¶
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# Install the library (uncomment if not already installed)
# !pip install encoding-atlas==0.2.0
# Install the library (uncomment if not already installed)
# !pip install encoding-atlas==0.2.0
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import numpy as np
import warnings
import math
import pickle
import logging
from itertools import combinations
print("NumPy version:", np.__version__)
import numpy as np
import warnings
import math
import pickle
import logging
from itertools import combinations
print("NumPy version:", np.__version__)
NumPy version: 2.2.6
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# Check which backends are available
backends_available = {}
try:
import pennylane as qml
backends_available["pennylane"] = qml.__version__
except ImportError:
backends_available["pennylane"] = None
try:
import qiskit
backends_available["qiskit"] = qiskit.__version__
except ImportError:
backends_available["qiskit"] = None
try:
import cirq
backends_available["cirq"] = cirq.__version__
except ImportError:
backends_available["cirq"] = None
for name, ver in backends_available.items():
status = f"v{ver}" if ver else "NOT INSTALLED"
print(f" {name:12s}: {status}")
# Check which backends are available
backends_available = {}
try:
import pennylane as qml
backends_available["pennylane"] = qml.__version__
except ImportError:
backends_available["pennylane"] = None
try:
import qiskit
backends_available["qiskit"] = qiskit.__version__
except ImportError:
backends_available["qiskit"] = None
try:
import cirq
backends_available["cirq"] = cirq.__version__
except ImportError:
backends_available["cirq"] = None
for name, ver in backends_available.items():
status = f"v{ver}" if ver else "NOT INSTALLED"
print(f" {name:12s}: {status}")
pennylane : v0.42.3 qiskit : v2.3.0 cirq : v1.5.0
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import encoding_atlas
print("encoding-atlas version:", encoding_atlas.__version__)
import encoding_atlas
print("encoding-atlas version:", encoding_atlas.__version__)
encoding-atlas version: 0.2.0
2. Constructor Parameters & Validation¶
HigherOrderAngleEncoding accepts the following parameters:
| Parameter | Type | Default | Description |
|---|---|---|---|
n_features |
int |
(required) | Number of classical input features (≥ 1) |
order |
int |
2 |
Maximum polynomial order (1 to n_features) |
rotation |
str |
'Y' |
Rotation axis: 'X', 'Y', or 'Z' |
combination |
str |
'product' |
Feature combination: 'product' or 'sum' |
include_first_order |
bool |
True |
Whether to include single-feature terms |
scaling |
float |
1.0 |
Scaling factor for all rotation angles |
reps |
int |
1 |
Number of circuit repetitions |
2.1 Default Construction¶
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from encoding_atlas import HigherOrderAngleEncoding
# Create with only the required parameter
enc = HigherOrderAngleEncoding(n_features=4)
print(f"n_features: {enc.n_features}")
print(f"order: {enc.order}")
print(f"rotation: {enc.rotation}")
print(f"combination: {enc.combination}")
print(f"include_first_order: {enc.include_first_order}")
print(f"scaling: {enc.scaling}")
print(f"reps: {enc.reps}")
from encoding_atlas import HigherOrderAngleEncoding
# Create with only the required parameter
enc = HigherOrderAngleEncoding(n_features=4)
print(f"n_features: {enc.n_features}")
print(f"order: {enc.order}")
print(f"rotation: {enc.rotation}")
print(f"combination: {enc.combination}")
print(f"include_first_order: {enc.include_first_order}")
print(f"scaling: {enc.scaling}")
print(f"reps: {enc.reps}")
n_features: 4 order: 2 rotation: Y combination: product include_first_order: True scaling: 1.0 reps: 1
2.2 Custom Construction¶
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# Fully customized encoding
enc_custom = HigherOrderAngleEncoding(
n_features=5,
order=3,
rotation="Z",
combination="sum",
include_first_order=False,
scaling=0.5,
reps=2,
)
print(f"n_features: {enc_custom.n_features}")
print(f"order: {enc_custom.order}")
print(f"rotation: {enc_custom.rotation}")
print(f"combination: {enc_custom.combination}")
print(f"include_first_order: {enc_custom.include_first_order}")
print(f"scaling: {enc_custom.scaling}")
print(f"reps: {enc_custom.reps}")
# Fully customized encoding
enc_custom = HigherOrderAngleEncoding(
n_features=5,
order=3,
rotation="Z",
combination="sum",
include_first_order=False,
scaling=0.5,
reps=2,
)
print(f"n_features: {enc_custom.n_features}")
print(f"order: {enc_custom.order}")
print(f"rotation: {enc_custom.rotation}")
print(f"combination: {enc_custom.combination}")
print(f"include_first_order: {enc_custom.include_first_order}")
print(f"scaling: {enc_custom.scaling}")
print(f"reps: {enc_custom.reps}")
n_features: 5 order: 3 rotation: Z combination: sum include_first_order: False scaling: 0.5 reps: 2
2.3 Input Normalisation¶
Rotation and combination strings are case-insensitive:
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# Case-insensitive: lowercase/mixed-case accepted and normalized
enc_lower = HigherOrderAngleEncoding(n_features=4, rotation="y", combination="PRODUCT")
print(f"rotation stored as: {enc_lower.rotation!r}") # 'Y'
print(f"combination stored as: {enc_lower.combination!r}") # 'product'
# Case-insensitive: lowercase/mixed-case accepted and normalized
enc_lower = HigherOrderAngleEncoding(n_features=4, rotation="y", combination="PRODUCT")
print(f"rotation stored as: {enc_lower.rotation!r}") # 'Y'
print(f"combination stored as: {enc_lower.combination!r}") # 'product'
rotation stored as: 'Y' combination stored as: 'product'
2.4 Parameter Validation¶
The constructor validates all parameters strictly. Let's demonstrate every validation error:
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# --- Invalid n_features ---
for bad_n in [0, -1]:
try:
HigherOrderAngleEncoding(n_features=bad_n)
except ValueError as e:
print(f"n_features={bad_n}: ValueError — {e}")
# Type errors for n_features
for bad_n in [4.0, "4", None]:
try:
HigherOrderAngleEncoding(n_features=bad_n)
except (TypeError, ValueError) as e:
print(f"n_features={bad_n!r}: {type(e).__name__} — {e}")
# --- Invalid n_features ---
for bad_n in [0, -1]:
try:
HigherOrderAngleEncoding(n_features=bad_n)
except ValueError as e:
print(f"n_features={bad_n}: ValueError — {e}")
# Type errors for n_features
for bad_n in [4.0, "4", None]:
try:
HigherOrderAngleEncoding(n_features=bad_n)
except (TypeError, ValueError) as e:
print(f"n_features={bad_n!r}: {type(e).__name__} — {e}")
n_features=0: ValueError — n_features must be at least 1, got 0 n_features=-1: ValueError — n_features must be at least 1, got -1 n_features=4.0: TypeError — n_features must be an integer, got float n_features='4': TypeError — n_features must be an integer, got str n_features=None: TypeError — n_features must be an integer, got NoneType
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# --- Invalid order ---
for bad_order in [0, -1]:
try:
HigherOrderAngleEncoding(n_features=4, order=bad_order)
except ValueError as e:
print(f"order={bad_order}: ValueError — {e}")
# order > n_features
try:
HigherOrderAngleEncoding(n_features=3, order=4)
except ValueError as e:
print(f"order=4 > n_features=3: ValueError — {e}")
# Float order
try:
HigherOrderAngleEncoding(n_features=4, order=2.5)
except TypeError as e:
print(f"order=2.5: TypeError — {e}")
# --- Invalid order ---
for bad_order in [0, -1]:
try:
HigherOrderAngleEncoding(n_features=4, order=bad_order)
except ValueError as e:
print(f"order={bad_order}: ValueError — {e}")
# order > n_features
try:
HigherOrderAngleEncoding(n_features=3, order=4)
except ValueError as e:
print(f"order=4 > n_features=3: ValueError — {e}")
# Float order
try:
HigherOrderAngleEncoding(n_features=4, order=2.5)
except TypeError as e:
print(f"order=2.5: TypeError — {e}")
order=0: ValueError — order must be at least 1, got 0 order=-1: ValueError — order must be at least 1, got -1 order=4 > n_features=3: ValueError — order (4) cannot exceed n_features (3) order=2.5: TypeError — order must be an integer, got float
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# --- Invalid rotation ---
try:
HigherOrderAngleEncoding(n_features=4, rotation="W")
except ValueError as e:
print(f"rotation='W': ValueError — {e}")
try:
HigherOrderAngleEncoding(n_features=4, rotation=1)
except TypeError as e:
print(f"rotation=1: TypeError — {e}")
# --- Invalid rotation ---
try:
HigherOrderAngleEncoding(n_features=4, rotation="W")
except ValueError as e:
print(f"rotation='W': ValueError — {e}")
try:
HigherOrderAngleEncoding(n_features=4, rotation=1)
except TypeError as e:
print(f"rotation=1: TypeError — {e}")
rotation='W': ValueError — rotation must be one of ['X', 'Y', 'Z'], got 'W' rotation=1: TypeError — rotation must be a string, got int
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# --- Invalid combination ---
try:
HigherOrderAngleEncoding(n_features=4, combination="multiply")
except ValueError as e:
print(f"combination='multiply': ValueError — {e}")
try:
HigherOrderAngleEncoding(n_features=4, combination=1)
except TypeError as e:
print(f"combination=1: TypeError — {e}")
# --- Invalid combination ---
try:
HigherOrderAngleEncoding(n_features=4, combination="multiply")
except ValueError as e:
print(f"combination='multiply': ValueError — {e}")
try:
HigherOrderAngleEncoding(n_features=4, combination=1)
except TypeError as e:
print(f"combination=1: TypeError — {e}")
combination='multiply': ValueError — combination must be one of ['product', 'sum'], got 'multiply' combination=1: TypeError — combination must be a string, got int
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# --- Invalid reps ---
for bad_reps in [0, -1]:
try:
HigherOrderAngleEncoding(n_features=4, reps=bad_reps)
except ValueError as e:
print(f"reps={bad_reps}: ValueError — {e}")
try:
HigherOrderAngleEncoding(n_features=4, reps=2.0)
except TypeError as e:
print(f"reps=2.0: TypeError — {e}")
# --- Invalid reps ---
for bad_reps in [0, -1]:
try:
HigherOrderAngleEncoding(n_features=4, reps=bad_reps)
except ValueError as e:
print(f"reps={bad_reps}: ValueError — {e}")
try:
HigherOrderAngleEncoding(n_features=4, reps=2.0)
except TypeError as e:
print(f"reps=2.0: TypeError — {e}")
reps=0: ValueError — reps must be at least 1, got 0 reps=-1: ValueError — reps must be at least 1, got -1 reps=2.0: TypeError — reps must be an integer, got float
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# --- Invalid scaling ---
try:
HigherOrderAngleEncoding(n_features=4, scaling=float("nan"))
except ValueError as e:
print(f"scaling=NaN: ValueError — {e}")
try:
HigherOrderAngleEncoding(n_features=4, scaling=float("inf"))
except ValueError as e:
print(f"scaling=inf: ValueError — {e}")
try:
HigherOrderAngleEncoding(n_features=4, scaling="1.0")
except TypeError as e:
print(f"scaling='1.0': TypeError — {e}")
# --- Invalid scaling ---
try:
HigherOrderAngleEncoding(n_features=4, scaling=float("nan"))
except ValueError as e:
print(f"scaling=NaN: ValueError — {e}")
try:
HigherOrderAngleEncoding(n_features=4, scaling=float("inf"))
except ValueError as e:
print(f"scaling=inf: ValueError — {e}")
try:
HigherOrderAngleEncoding(n_features=4, scaling="1.0")
except TypeError as e:
print(f"scaling='1.0': TypeError — {e}")
scaling=NaN: ValueError — scaling must be finite, got nan scaling=inf: ValueError — scaling must be finite, got inf scaling='1.0': TypeError — scaling must be a number, got str
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# --- Invalid include_first_order ---
try:
HigherOrderAngleEncoding(n_features=4, include_first_order="True")
except TypeError as e:
print(f"include_first_order='True': TypeError — {e}")
# --- Invalid include_first_order ---
try:
HigherOrderAngleEncoding(n_features=4, include_first_order="True")
except TypeError as e:
print(f"include_first_order='True': TypeError — {e}")
include_first_order='True': TypeError — include_first_order must be a bool, got str
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# --- Degenerate case: order=1 with include_first_order=False ---
# This would produce zero terms, which is meaningless
try:
HigherOrderAngleEncoding(n_features=4, order=1, include_first_order=False)
except ValueError as e:
print(f"Degenerate case: ValueError — {e}")
# --- Degenerate case: order=1 with include_first_order=False ---
# This would produce zero terms, which is meaningless
try:
HigherOrderAngleEncoding(n_features=4, order=1, include_first_order=False)
except ValueError as e:
print(f"Degenerate case: ValueError — {e}")
Degenerate case: ValueError — Cannot have order=1 with include_first_order=False (would result in no terms)
2.5 Input Data Validation¶
The get_circuit() method validates input data thoroughly:
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enc = HigherOrderAngleEncoding(n_features=4)
x_good = np.array([0.1, 0.2, 0.3, 0.4])
# Wrong number of features
try:
enc.get_circuit(np.array([0.1, 0.2, 0.3]))
except ValueError as e:
print(f"Wrong features: {e}")
# NaN values
try:
enc.get_circuit(np.array([0.1, np.nan, 0.3, 0.4]))
except ValueError as e:
print(f"NaN input: {e}")
# Infinite values
try:
enc.get_circuit(np.array([0.1, np.inf, 0.3, 0.4]))
except ValueError as e:
print(f"Inf input: {e}")
# Multiple samples passed to get_circuit (use get_circuits instead)
try:
enc.get_circuit(np.array([[0.1, 0.2, 0.3, 0.4], [0.5, 0.6, 0.7, 0.8]]))
except ValueError as e:
print(f"Batch to get_circuit: {e}")
# List input is accepted (auto-converted)
circuit = enc.get_circuit([0.1, 0.2, 0.3, 0.4])
print(f"\nList input accepted: {callable(circuit)}")
# 2D single-sample input is accepted
circuit = enc.get_circuit(np.array([[0.1, 0.2, 0.3, 0.4]]))
print(f"2D single-sample accepted: {callable(circuit)}")
enc = HigherOrderAngleEncoding(n_features=4)
x_good = np.array([0.1, 0.2, 0.3, 0.4])
# Wrong number of features
try:
enc.get_circuit(np.array([0.1, 0.2, 0.3]))
except ValueError as e:
print(f"Wrong features: {e}")
# NaN values
try:
enc.get_circuit(np.array([0.1, np.nan, 0.3, 0.4]))
except ValueError as e:
print(f"NaN input: {e}")
# Infinite values
try:
enc.get_circuit(np.array([0.1, np.inf, 0.3, 0.4]))
except ValueError as e:
print(f"Inf input: {e}")
# Multiple samples passed to get_circuit (use get_circuits instead)
try:
enc.get_circuit(np.array([[0.1, 0.2, 0.3, 0.4], [0.5, 0.6, 0.7, 0.8]]))
except ValueError as e:
print(f"Batch to get_circuit: {e}")
# List input is accepted (auto-converted)
circuit = enc.get_circuit([0.1, 0.2, 0.3, 0.4])
print(f"\nList input accepted: {callable(circuit)}")
# 2D single-sample input is accepted
circuit = enc.get_circuit(np.array([[0.1, 0.2, 0.3, 0.4]]))
print(f"2D single-sample accepted: {callable(circuit)}")
Wrong features: Expected 4 features, got 3 NaN input: Input contains NaN or infinite values Inf input: Input contains NaN or infinite values Batch to get_circuit: get_circuit expects a single sample, got shape (2, 4). Use get_circuits for batches. List input accepted: True 2D single-sample accepted: True
3. Core Properties¶
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enc = HigherOrderAngleEncoding(n_features=4, order=2, reps=2)
print(f"n_features: {enc.n_features}")
print(f"n_qubits: {enc.n_qubits} (always equals n_features)")
print(f"depth: {enc.depth} (equals reps, since all gates are parallel)")
print(f"n_terms: {enc.n_terms} (total polynomial terms)")
print(f"order: {enc.order}")
print(f"rotation: {enc.rotation}")
print(f"combination:{enc.combination}")
print(f"scaling: {enc.scaling}")
print(f"reps: {enc.reps}")
enc = HigherOrderAngleEncoding(n_features=4, order=2, reps=2)
print(f"n_features: {enc.n_features}")
print(f"n_qubits: {enc.n_qubits} (always equals n_features)")
print(f"depth: {enc.depth} (equals reps, since all gates are parallel)")
print(f"n_terms: {enc.n_terms} (total polynomial terms)")
print(f"order: {enc.order}")
print(f"rotation: {enc.rotation}")
print(f"combination:{enc.combination}")
print(f"scaling: {enc.scaling}")
print(f"reps: {enc.reps}")
n_features: 4 n_qubits: 4 (always equals n_features) depth: 2 (equals reps, since all gates are parallel) n_terms: 10 (total polynomial terms) order: 2 rotation: Y combination:product scaling: 1.0 reps: 2
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# The config property returns a copy of the encoding-specific parameters
config = enc.config
print("Config:", config)
# Verify it's a copy (modifying it doesn't affect the encoding)
config["order"] = 999
print(f"\nOriginal order unchanged: {enc.order}")
# The config property returns a copy of the encoding-specific parameters
config = enc.config
print("Config:", config)
# Verify it's a copy (modifying it doesn't affect the encoding)
config["order"] = 999
print(f"\nOriginal order unchanged: {enc.order}")
Config: {'order': 2, 'rotation': 'Y', 'combination': 'product', 'include_first_order': True, 'scaling': 1.0, 'reps': 2}
Original order unchanged: 2
4. Polynomial Term Generation & Assignment¶
Terms are generated in lexicographic order by degree, then assigned to qubits using round-robin by term index:
$$\text{qubit\_idx} = \text{term\_index} \mod n\_\text{qubits}$$
4.1 Viewing Terms¶
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enc = HigherOrderAngleEncoding(n_features=3, order=2)
print("All terms:")
for i, term in enumerate(enc.terms):
order = len(term)
label = " × ".join(f"x_{j}" for j in term)
print(f" Index {i}: {str(term):10s} → {label:15s} (order {order})")
print(f"\nTotal terms: {enc.n_terms}")
enc = HigherOrderAngleEncoding(n_features=3, order=2)
print("All terms:")
for i, term in enumerate(enc.terms):
order = len(term)
label = " × ".join(f"x_{j}" for j in term)
print(f" Index {i}: {str(term):10s} → {label:15s} (order {order})")
print(f"\nTotal terms: {enc.n_terms}")
All terms: Index 0: (0,) → x_0 (order 1) Index 1: (1,) → x_1 (order 1) Index 2: (2,) → x_2 (order 1) Index 3: (0, 1) → x_0 × x_1 (order 2) Index 4: (0, 2) → x_0 × x_2 (order 2) Index 5: (1, 2) → x_1 × x_2 (order 2) Total terms: 6
4.2 Detailed Term Info¶
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info = enc.get_term_info()
print("Terms by order:")
for order, terms in sorted(info["terms_by_order"].items()):
print(f" Order {order}: {terms}")
print("\nQubit assignments (round-robin):")
for qubit, terms in sorted(info["qubit_assignments"].items()):
labels = [" × ".join(f"x_{j}" for j in t) for t in terms]
print(f" Qubit {qubit}: {terms} → [{', '.join(labels)}]")
info = enc.get_term_info()
print("Terms by order:")
for order, terms in sorted(info["terms_by_order"].items()):
print(f" Order {order}: {terms}")
print("\nQubit assignments (round-robin):")
for qubit, terms in sorted(info["qubit_assignments"].items()):
labels = [" × ".join(f"x_{j}" for j in t) for t in terms]
print(f" Qubit {qubit}: {terms} → [{', '.join(labels)}]")
Terms by order: Order 1: [(0,), (1,), (2,)] Order 2: [(0, 1), (0, 2), (1, 2)] Qubit assignments (round-robin): Qubit 0: [(0,), (0, 1)] → [x_0, x_0 × x_1] Qubit 1: [(1,), (0, 2)] → [x_1, x_0 × x_2] Qubit 2: [(2,), (1, 2)] → [x_2, x_1 × x_2]
4.3 Third-Order Terms¶
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enc3 = HigherOrderAngleEncoding(n_features=4, order=3)
info3 = enc3.get_term_info()
print(f"Total terms: {enc3.n_terms}")
print(f" Order 1: {len(info3['terms_by_order'][1])} terms")
print(f" Order 2: {len(info3['terms_by_order'][2])} terms")
print(f" Order 3: {len(info3['terms_by_order'][3])} terms")
print("\nThird-order terms:")
for t in info3["terms_by_order"][3]:
print(f" {t} → " + " × ".join(f"x_{j}" for j in t))
enc3 = HigherOrderAngleEncoding(n_features=4, order=3)
info3 = enc3.get_term_info()
print(f"Total terms: {enc3.n_terms}")
print(f" Order 1: {len(info3['terms_by_order'][1])} terms")
print(f" Order 2: {len(info3['terms_by_order'][2])} terms")
print(f" Order 3: {len(info3['terms_by_order'][3])} terms")
print("\nThird-order terms:")
for t in info3["terms_by_order"][3]:
print(f" {t} → " + " × ".join(f"x_{j}" for j in t))
Total terms: 14 Order 1: 4 terms Order 2: 6 terms Order 3: 4 terms Third-order terms: (0, 1, 2) → x_0 × x_1 × x_2 (0, 1, 3) → x_0 × x_1 × x_3 (0, 2, 3) → x_0 × x_2 × x_3 (1, 2, 3) → x_1 × x_2 × x_3
4.4 Maximum Order (all subsets)¶
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enc_max = HigherOrderAngleEncoding(n_features=4, order=4)
print(f"n_features=4, order=4 → {enc_max.n_terms} terms (2^4 - 1 = {2**4 - 1})")
# Verify: total non-empty subsets = 2^n - 1
assert enc_max.n_terms == 2**4 - 1
enc_max = HigherOrderAngleEncoding(n_features=4, order=4)
print(f"n_features=4, order=4 → {enc_max.n_terms} terms (2^4 - 1 = {2**4 - 1})")
# Verify: total non-empty subsets = 2^n - 1
assert enc_max.n_terms == 2**4 - 1
n_features=4, order=4 → 15 terms (2^4 - 1 = 15)
4.5 Without First-Order Terms¶
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enc_no_first = HigherOrderAngleEncoding(n_features=4, order=2, include_first_order=False)
print("Terms (no first-order):")
for i, term in enumerate(enc_no_first.terms):
print(f" Index {i}: {term} → " + " × ".join(f"x_{j}" for j in term))
print(f"\nTotal: {enc_no_first.n_terms} terms (only pairwise)")
enc_no_first = HigherOrderAngleEncoding(n_features=4, order=2, include_first_order=False)
print("Terms (no first-order):")
for i, term in enumerate(enc_no_first.terms):
print(f" Index {i}: {term} → " + " × ".join(f"x_{j}" for j in term))
print(f"\nTotal: {enc_no_first.n_terms} terms (only pairwise)")
Terms (no first-order): Index 0: (0, 1) → x_0 × x_1 Index 1: (0, 2) → x_0 × x_2 Index 2: (0, 3) → x_0 × x_3 Index 3: (1, 2) → x_1 × x_2 Index 4: (1, 3) → x_1 × x_3 Index 5: (2, 3) → x_2 × x_3 Total: 6 terms (only pairwise)
5. Rotation Angle Computation¶
The compute_angles() method reveals the rotation angle each qubit will receive for a given input.
5.1 Product Combination¶
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enc = HigherOrderAngleEncoding(n_features=3, order=2, combination="product", scaling=1.0)
x = np.array([2.0, 3.0, 5.0])
angles = enc.compute_angles(x)
# Manual verification:
# Terms: (0,), (1,), (2,), (0,1), (0,2), (1,2)
# Values: 2, 3, 5, 6, 10, 15
# Assignment (round-robin with 3 qubits):
# Qubit 0: indices 0, 3 → values 2 + 6 = 8
# Qubit 1: indices 1, 4 → values 3 + 10 = 13
# Qubit 2: indices 2, 5 → values 5 + 15 = 20
print("Input:", x)
print("Computed angles:", angles)
print("Expected: [8.0, 13.0, 20.0]")
assert np.allclose(angles, [8.0, 13.0, 20.0])
enc = HigherOrderAngleEncoding(n_features=3, order=2, combination="product", scaling=1.0)
x = np.array([2.0, 3.0, 5.0])
angles = enc.compute_angles(x)
# Manual verification:
# Terms: (0,), (1,), (2,), (0,1), (0,2), (1,2)
# Values: 2, 3, 5, 6, 10, 15
# Assignment (round-robin with 3 qubits):
# Qubit 0: indices 0, 3 → values 2 + 6 = 8
# Qubit 1: indices 1, 4 → values 3 + 10 = 13
# Qubit 2: indices 2, 5 → values 5 + 15 = 20
print("Input:", x)
print("Computed angles:", angles)
print("Expected: [8.0, 13.0, 20.0]")
assert np.allclose(angles, [8.0, 13.0, 20.0])
Input: [2. 3. 5.] Computed angles: [ 8. 13. 20.] Expected: [8.0, 13.0, 20.0]
5.2 Sum Combination¶
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enc_sum = HigherOrderAngleEncoding(n_features=3, order=2, combination="sum", scaling=1.0)
x = np.array([2.0, 3.0, 5.0])
angles_sum = enc_sum.compute_angles(x)
# With sum combination, term values are SUMS instead of products:
# Terms: (0,), (1,), (2,), (0,1), (0,2), (1,2)
# Values: 2, 3, 5, 5, 7, 8
# Qubit 0: 2 + 5 = 7
# Qubit 1: 3 + 7 = 10
# Qubit 2: 5 + 8 = 13
print("Sum combination angles:", angles_sum)
print("Expected: [7.0, 10.0, 13.0]")
assert np.allclose(angles_sum, [7.0, 10.0, 13.0])
enc_sum = HigherOrderAngleEncoding(n_features=3, order=2, combination="sum", scaling=1.0)
x = np.array([2.0, 3.0, 5.0])
angles_sum = enc_sum.compute_angles(x)
# With sum combination, term values are SUMS instead of products:
# Terms: (0,), (1,), (2,), (0,1), (0,2), (1,2)
# Values: 2, 3, 5, 5, 7, 8
# Qubit 0: 2 + 5 = 7
# Qubit 1: 3 + 7 = 10
# Qubit 2: 5 + 8 = 13
print("Sum combination angles:", angles_sum)
print("Expected: [7.0, 10.0, 13.0]")
assert np.allclose(angles_sum, [7.0, 10.0, 13.0])
Sum combination angles: [ 7. 10. 13.] Expected: [7.0, 10.0, 13.0]
5.3 Scaling Effect¶
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enc_s1 = HigherOrderAngleEncoding(n_features=3, order=1, scaling=1.0)
enc_s2 = HigherOrderAngleEncoding(n_features=3, order=1, scaling=2.0)
x = np.array([0.1, 0.2, 0.3])
angles_s1 = enc_s1.compute_angles(x)
angles_s2 = enc_s2.compute_angles(x)
print(f"scaling=1.0: {angles_s1}")
print(f"scaling=2.0: {angles_s2}")
print(f"Ratio: {angles_s2 / angles_s1}")
assert np.allclose(angles_s2, 2.0 * angles_s1)
enc_s1 = HigherOrderAngleEncoding(n_features=3, order=1, scaling=1.0)
enc_s2 = HigherOrderAngleEncoding(n_features=3, order=1, scaling=2.0)
x = np.array([0.1, 0.2, 0.3])
angles_s1 = enc_s1.compute_angles(x)
angles_s2 = enc_s2.compute_angles(x)
print(f"scaling=1.0: {angles_s1}")
print(f"scaling=2.0: {angles_s2}")
print(f"Ratio: {angles_s2 / angles_s1}")
assert np.allclose(angles_s2, 2.0 * angles_s1)
scaling=1.0: [0.1 0.2 0.3] scaling=2.0: [0.2 0.4 0.6] Ratio: [2. 2. 2.]
5.4 First-Order Only = Standard Angle Encoding¶
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enc_order1 = HigherOrderAngleEncoding(n_features=4, order=1, scaling=1.0)
x = np.array([0.1, 0.2, 0.3, 0.4])
angles = enc_order1.compute_angles(x)
print(f"Order-1 angles: {angles}")
print(f"Input features: {x}")
print(f"Identical: {np.allclose(angles, x)}")
enc_order1 = HigherOrderAngleEncoding(n_features=4, order=1, scaling=1.0)
x = np.array([0.1, 0.2, 0.3, 0.4])
angles = enc_order1.compute_angles(x)
print(f"Order-1 angles: {angles}")
print(f"Input features: {x}")
print(f"Identical: {np.allclose(angles, x)}")
Order-1 angles: [0.1 0.2 0.3 0.4] Input features: [0.1 0.2 0.3 0.4] Identical: True
5.5 Third-Order Product Verification¶
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enc3 = HigherOrderAngleEncoding(n_features=3, order=3, combination="product", scaling=1.0)
x = np.array([2.0, 3.0, 5.0])
# Terms: (0,), (1,), (2,), (0,1), (0,2), (1,2), (0,1,2)
# Values: 2, 3, 5, 6, 10, 15, 30
# Qubit 0: indices 0, 3, 6 → 2 + 6 + 30 = 38
# Qubit 1: indices 1, 4 → 3 + 10 = 13
# Qubit 2: indices 2, 5 → 5 + 15 = 20
angles = enc3.compute_angles(x)
print(f"Third-order angles: {angles}")
print(f"Expected: [38.0, 13.0, 20.0]")
assert np.allclose(angles, [38.0, 13.0, 20.0])
enc3 = HigherOrderAngleEncoding(n_features=3, order=3, combination="product", scaling=1.0)
x = np.array([2.0, 3.0, 5.0])
# Terms: (0,), (1,), (2,), (0,1), (0,2), (1,2), (0,1,2)
# Values: 2, 3, 5, 6, 10, 15, 30
# Qubit 0: indices 0, 3, 6 → 2 + 6 + 30 = 38
# Qubit 1: indices 1, 4 → 3 + 10 = 13
# Qubit 2: indices 2, 5 → 5 + 15 = 20
angles = enc3.compute_angles(x)
print(f"Third-order angles: {angles}")
print(f"Expected: [38.0, 13.0, 20.0]")
assert np.allclose(angles, [38.0, 13.0, 20.0])
Third-order angles: [38. 13. 20.] Expected: [38.0, 13.0, 20.0]
6. Circuit Generation — PennyLane Backend¶
The PennyLane backend returns a callable function that applies rotation gates when called inside a QNode.
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enc = HigherOrderAngleEncoding(n_features=4, order=2)
x = np.array([0.1, 0.2, 0.3, 0.4])
circuit_fn = enc.get_circuit(x, backend="pennylane")
print(f"Type: {type(circuit_fn)}")
print(f"Callable: {callable(circuit_fn)}")
enc = HigherOrderAngleEncoding(n_features=4, order=2)
x = np.array([0.1, 0.2, 0.3, 0.4])
circuit_fn = enc.get_circuit(x, backend="pennylane")
print(f"Type: {type(circuit_fn)}")
print(f"Callable: {callable(circuit_fn)}")
Type: <class 'function'> Callable: True
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# Execute inside a QNode to get the statevector
import pennylane as qml
dev = qml.device("default.qubit", wires=enc.n_qubits)
@qml.qnode(dev)
def full_circuit():
circuit_fn()
return qml.state()
state = full_circuit()
print(f"State vector shape: {state.shape}")
print(f"State vector norm: {np.sum(np.abs(state)**2):.10f}")
print(f"First 4 amplitudes: {state[:4]}")
# Execute inside a QNode to get the statevector
import pennylane as qml
dev = qml.device("default.qubit", wires=enc.n_qubits)
@qml.qnode(dev)
def full_circuit():
circuit_fn()
return qml.state()
state = full_circuit()
print(f"State vector shape: {state.shape}")
print(f"State vector norm: {np.sum(np.abs(state)**2):.10f}")
print(f"First 4 amplitudes: {state[:4]}")
State vector shape: (16,) State vector norm: 1.0000000000 First 4 amplitudes: [0.94025295+0.j 0.22015399+0.j 0.16140083+0.j 0.03779093+0.j]
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# Visualize the circuit
@qml.qnode(dev)
def draw_circuit():
circuit_fn()
return qml.state()
print(qml.draw(draw_circuit)())
# Visualize the circuit
@qml.qnode(dev)
def draw_circuit():
circuit_fn()
return qml.state()
print(qml.draw(draw_circuit)())
0: ──RY(0.20)─┤ State 1: ──RY(0.35)─┤ State 2: ──RY(0.34)─┤ State 3: ──RY(0.46)─┤ State
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# With multiple reps
enc_reps = HigherOrderAngleEncoding(n_features=4, order=2, reps=2)
circuit_fn_reps = enc_reps.get_circuit(x, backend="pennylane")
dev2 = qml.device("default.qubit", wires=enc_reps.n_qubits)
@qml.qnode(dev2)
def draw_reps():
circuit_fn_reps()
return qml.state()
print("With reps=2:")
print(qml.draw(draw_reps)())
# With multiple reps
enc_reps = HigherOrderAngleEncoding(n_features=4, order=2, reps=2)
circuit_fn_reps = enc_reps.get_circuit(x, backend="pennylane")
dev2 = qml.device("default.qubit", wires=enc_reps.n_qubits)
@qml.qnode(dev2)
def draw_reps():
circuit_fn_reps()
return qml.state()
print("With reps=2:")
print(qml.draw(draw_reps)())
With reps=2: 0: ──RY(0.20)──RY(0.20)─┤ State 1: ──RY(0.35)──RY(0.35)─┤ State 2: ──RY(0.34)──RY(0.34)─┤ State 3: ──RY(0.46)──RY(0.46)─┤ State
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# Different rotation axes
for rot in ["X", "Y", "Z"]:
enc_rot = HigherOrderAngleEncoding(n_features=3, order=1, rotation=rot)
fn = enc_rot.get_circuit(np.array([0.5, 1.0, 1.5]), backend="pennylane")
d = qml.device("default.qubit", wires=3)
@qml.qnode(d)
def show():
fn()
return qml.state()
print(f"rotation='{rot}':")
print(qml.draw(show)())
print()
# Different rotation axes
for rot in ["X", "Y", "Z"]:
enc_rot = HigherOrderAngleEncoding(n_features=3, order=1, rotation=rot)
fn = enc_rot.get_circuit(np.array([0.5, 1.0, 1.5]), backend="pennylane")
d = qml.device("default.qubit", wires=3)
@qml.qnode(d)
def show():
fn()
return qml.state()
print(f"rotation='{rot}':")
print(qml.draw(show)())
print()
rotation='X': 0: ──RX(0.50)─┤ State 1: ──RX(1.00)─┤ State 2: ──RX(1.50)─┤ State rotation='Y': 0: ──RY(0.50)─┤ State 1: ──RY(1.00)─┤ State 2: ──RY(1.50)─┤ State rotation='Z': 0: ──RZ(0.50)─┤ State 1: ──RZ(1.00)─┤ State 2: ──RZ(1.50)─┤ State
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# Backend name is case-insensitive and None defaults to PennyLane
c1 = enc.get_circuit(x, backend="pennylane")
c2 = enc.get_circuit(x, backend="PENNYLANE")
c3 = enc.get_circuit(x, backend="PennyLane")
c4 = enc.get_circuit(x, backend=None)
print(f"All callables: {all(callable(c) for c in [c1, c2, c3, c4])}")
# Backend name is case-insensitive and None defaults to PennyLane
c1 = enc.get_circuit(x, backend="pennylane")
c2 = enc.get_circuit(x, backend="PENNYLANE")
c3 = enc.get_circuit(x, backend="PennyLane")
c4 = enc.get_circuit(x, backend=None)
print(f"All callables: {all(callable(c) for c in [c1, c2, c3, c4])}")
All callables: True
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from qiskit import QuantumCircuit
enc = HigherOrderAngleEncoding(n_features=4, order=2)
x = np.array([0.1, 0.2, 0.3, 0.4])
qc = enc.get_circuit(x, backend="qiskit")
print(f"Type: {type(qc).__name__}")
print(f"Num qubits: {qc.num_qubits}")
print(f"Depth: {qc.depth()}")
print(f"Gate count: {len(qc.data)}")
print()
print(qc.draw(output="text"))
from qiskit import QuantumCircuit
enc = HigherOrderAngleEncoding(n_features=4, order=2)
x = np.array([0.1, 0.2, 0.3, 0.4])
qc = enc.get_circuit(x, backend="qiskit")
print(f"Type: {type(qc).__name__}")
print(f"Num qubits: {qc.num_qubits}")
print(f"Depth: {qc.depth()}")
print(f"Gate count: {len(qc.data)}")
print()
print(qc.draw(output="text"))
Type: QuantumCircuit
Num qubits: 4
Depth: 1
Gate count: 4
┌─────────┐
q_0: ┤ Ry(0.2) ├─
├─────────┴┐
q_1: ┤ Ry(0.35) ├
├──────────┤
q_2: ┤ Ry(0.34) ├
├──────────┤
q_3: ┤ Ry(0.46) ├
└──────────┘
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# Verify gate types — default rotation='Y' → all ry gates
for instruction in qc.data:
if instruction.operation.name != "barrier":
assert instruction.operation.name == "ry", f"Unexpected gate: {instruction.operation.name}"
print("All gates are RY (as expected for rotation='Y')")
# Verify gate types — default rotation='Y' → all ry gates
for instruction in qc.data:
if instruction.operation.name != "barrier":
assert instruction.operation.name == "ry", f"Unexpected gate: {instruction.operation.name}"
print("All gates are RY (as expected for rotation='Y')")
All gates are RY (as expected for rotation='Y')
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# With multiple reps — barriers separate repetitions
enc_reps = HigherOrderAngleEncoding(n_features=4, order=2, reps=3)
qc_reps = enc_reps.get_circuit(x, backend="qiskit")
barrier_count = sum(1 for inst in qc_reps.data if inst.operation.name == "barrier")
print(f"Reps: {enc_reps.reps}, Barriers: {barrier_count} (reps - 1)")
print()
print(qc_reps.draw(output="text"))
# With multiple reps — barriers separate repetitions
enc_reps = HigherOrderAngleEncoding(n_features=4, order=2, reps=3)
qc_reps = enc_reps.get_circuit(x, backend="qiskit")
barrier_count = sum(1 for inst in qc_reps.data if inst.operation.name == "barrier")
print(f"Reps: {enc_reps.reps}, Barriers: {barrier_count} (reps - 1)")
print()
print(qc_reps.draw(output="text"))
Reps: 3, Barriers: 2 (reps - 1)
┌─────────┐ ░ ┌─────────┐ ░ ┌─────────┐
q_0: ┤ Ry(0.2) ├──░─┤ Ry(0.2) ├──░─┤ Ry(0.2) ├─
├─────────┴┐ ░ ├─────────┴┐ ░ ├─────────┴┐
q_1: ┤ Ry(0.35) ├─░─┤ Ry(0.35) ├─░─┤ Ry(0.35) ├
├──────────┤ ░ ├──────────┤ ░ ├──────────┤
q_2: ┤ Ry(0.34) ├─░─┤ Ry(0.34) ├─░─┤ Ry(0.34) ├
├──────────┤ ░ ├──────────┤ ░ ├──────────┤
q_3: ┤ Ry(0.46) ├─░─┤ Ry(0.46) ├─░─┤ Ry(0.46) ├
└──────────┘ ░ └──────────┘ ░ └──────────┘
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# Qiskit statevector simulation
from qiskit.quantum_info import Statevector
sv = Statevector.from_instruction(qc)
print(f"Statevector length: {len(sv)}")
print(f"Norm: {np.sum(np.abs(np.array(sv))**2):.10f}")
# Qiskit statevector simulation
from qiskit.quantum_info import Statevector
sv = Statevector.from_instruction(qc)
print(f"Statevector length: {len(sv)}")
print(f"Norm: {np.sum(np.abs(np.array(sv))**2):.10f}")
Statevector length: 16 Norm: 1.0000000000
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# RX and RZ gate verification
for rot in ["X", "Z"]:
enc_r = HigherOrderAngleEncoding(n_features=4, order=2, rotation=rot)
qc_r = enc_r.get_circuit(x, backend="qiskit")
gate_names = {inst.operation.name for inst in qc_r.data if inst.operation.name != "barrier"}
expected = f"r{rot.lower()}"
print(f"rotation='{rot}' → gates: {gate_names} (expected: {{'{expected}'}})")
assert gate_names == {expected}
# RX and RZ gate verification
for rot in ["X", "Z"]:
enc_r = HigherOrderAngleEncoding(n_features=4, order=2, rotation=rot)
qc_r = enc_r.get_circuit(x, backend="qiskit")
gate_names = {inst.operation.name for inst in qc_r.data if inst.operation.name != "barrier"}
expected = f"r{rot.lower()}"
print(f"rotation='{rot}' → gates: {gate_names} (expected: {{'{expected}'}})")
assert gate_names == {expected}
rotation='X' → gates: {'rx'} (expected: {'rx'})
rotation='Z' → gates: {'rz'} (expected: {'rz'})
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import cirq
enc = HigherOrderAngleEncoding(n_features=4, order=2)
x = np.array([0.1, 0.2, 0.3, 0.4])
cirq_circuit = enc.get_circuit(x, backend="cirq")
print(f"Type: {type(cirq_circuit).__name__}")
print(f"Num qubits: {len(cirq_circuit.all_qubits())}")
print(f"Moments: {len(cirq_circuit.moments)}")
print()
print(cirq_circuit)
import cirq
enc = HigherOrderAngleEncoding(n_features=4, order=2)
x = np.array([0.1, 0.2, 0.3, 0.4])
cirq_circuit = enc.get_circuit(x, backend="cirq")
print(f"Type: {type(cirq_circuit).__name__}")
print(f"Num qubits: {len(cirq_circuit.all_qubits())}")
print(f"Moments: {len(cirq_circuit.moments)}")
print()
print(cirq_circuit)
Type: Circuit Num qubits: 4 Moments: 1 0: ───Ry(0.064π)─── 1: ───Ry(0.111π)─── 2: ───Ry(0.108π)─── 3: ───Ry(0.146π)───
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# Cirq statevector simulation
simulator = cirq.Simulator()
result = simulator.simulate(cirq_circuit)
state_cirq = result.final_state_vector
print(f"Statevector length: {len(state_cirq)}")
print(f"Norm: {np.sum(np.abs(state_cirq)**2):.10f}")
# Cirq statevector simulation
simulator = cirq.Simulator()
result = simulator.simulate(cirq_circuit)
state_cirq = result.final_state_vector
print(f"Statevector length: {len(state_cirq)}")
print(f"Norm: {np.sum(np.abs(state_cirq)**2):.10f}")
Statevector length: 16 Norm: 1.0000001192
9. Batch Circuit Generation & Parallel Processing¶
get_circuits() generates circuits for multiple input samples at once.
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enc = HigherOrderAngleEncoding(n_features=4, order=2)
# Batch of 5 samples
X_batch = np.array([
[0.1, 0.2, 0.3, 0.4],
[0.5, 0.6, 0.7, 0.8],
[0.0, 0.0, 0.0, 0.0],
[-1.0, -2.0, 1.0, 2.0],
[np.pi, np.e, 1.0, 0.5],
])
# Sequential (default)
circuits_seq = enc.get_circuits(X_batch, backend="pennylane")
print(f"Sequential: {len(circuits_seq)} circuits, all callable: {all(callable(c) for c in circuits_seq)}")
# Parallel processing
circuits_par = enc.get_circuits(X_batch, backend="pennylane", parallel=True)
print(f"Parallel: {len(circuits_par)} circuits, all callable: {all(callable(c) for c in circuits_par)}")
# Parallel with custom max_workers
circuits_par2 = enc.get_circuits(X_batch, backend="pennylane", parallel=True, max_workers=2)
print(f"Parallel (2 workers): {len(circuits_par2)} circuits")
enc = HigherOrderAngleEncoding(n_features=4, order=2)
# Batch of 5 samples
X_batch = np.array([
[0.1, 0.2, 0.3, 0.4],
[0.5, 0.6, 0.7, 0.8],
[0.0, 0.0, 0.0, 0.0],
[-1.0, -2.0, 1.0, 2.0],
[np.pi, np.e, 1.0, 0.5],
])
# Sequential (default)
circuits_seq = enc.get_circuits(X_batch, backend="pennylane")
print(f"Sequential: {len(circuits_seq)} circuits, all callable: {all(callable(c) for c in circuits_seq)}")
# Parallel processing
circuits_par = enc.get_circuits(X_batch, backend="pennylane", parallel=True)
print(f"Parallel: {len(circuits_par)} circuits, all callable: {all(callable(c) for c in circuits_par)}")
# Parallel with custom max_workers
circuits_par2 = enc.get_circuits(X_batch, backend="pennylane", parallel=True, max_workers=2)
print(f"Parallel (2 workers): {len(circuits_par2)} circuits")
Sequential: 5 circuits, all callable: True Parallel: 5 circuits, all callable: True Parallel (2 workers): 5 circuits
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# 1D input is treated as a single sample
circuits_1d = enc.get_circuits(np.array([0.1, 0.2, 0.3, 0.4]), backend="pennylane")
print(f"1D input → {len(circuits_1d)} circuit(s)")
# 1D input is treated as a single sample
circuits_1d = enc.get_circuits(np.array([0.1, 0.2, 0.3, 0.4]), backend="pennylane")
print(f"1D input → {len(circuits_1d)} circuit(s)")
1D input → 1 circuit(s)
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# Batch with Qiskit backend
circuits_qiskit = enc.get_circuits(X_batch, backend="qiskit")
print(f"Qiskit batch: {len(circuits_qiskit)} circuits")
for i, qc in enumerate(circuits_qiskit):
print(f" Circuit {i}: {qc.num_qubits} qubits, {len(qc.data)} gates")
# Batch with Qiskit backend
circuits_qiskit = enc.get_circuits(X_batch, backend="qiskit")
print(f"Qiskit batch: {len(circuits_qiskit)} circuits")
for i, qc in enumerate(circuits_qiskit):
print(f" Circuit {i}: {qc.num_qubits} qubits, {len(qc.data)} gates")
Qiskit batch: 5 circuits Circuit 0: 4 qubits, 4 gates Circuit 1: 4 qubits, 4 gates Circuit 2: 4 qubits, 4 gates Circuit 3: 4 qubits, 4 gates Circuit 4: 4 qubits, 4 gates
10. Gate Count Breakdown¶
gate_count_breakdown() returns a typed dictionary with detailed gate counts.
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enc = HigherOrderAngleEncoding(n_features=4, order=2, rotation="Y", reps=2)
breakdown = enc.gate_count_breakdown()
print("Gate Count Breakdown:")
for key, value in breakdown.items():
print(f" {key:20s}: {value}")
enc = HigherOrderAngleEncoding(n_features=4, order=2, rotation="Y", reps=2)
breakdown = enc.gate_count_breakdown()
print("Gate Count Breakdown:")
for key, value in breakdown.items():
print(f" {key:20s}: {value}")
Gate Count Breakdown: rx : 0 ry : 8 rz : 0 total_single_qubit : 8 total_two_qubit : 0 total : 8
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# Only the configured rotation gate has non-zero count
for rot in ["X", "Y", "Z"]:
enc_r = HigherOrderAngleEncoding(n_features=4, order=2, rotation=rot)
bd = enc_r.gate_count_breakdown()
print(f"rotation='{rot}': rx={bd['rx']}, ry={bd['ry']}, rz={bd['rz']}, total={bd['total']}")
# Only the configured rotation gate has non-zero count
for rot in ["X", "Y", "Z"]:
enc_r = HigherOrderAngleEncoding(n_features=4, order=2, rotation=rot)
bd = enc_r.gate_count_breakdown()
print(f"rotation='{rot}': rx={bd['rx']}, ry={bd['ry']}, rz={bd['rz']}, total={bd['total']}")
rotation='X': rx=4, ry=0, rz=0, total=4 rotation='Y': rx=0, ry=4, rz=0, total=4 rotation='Z': rx=0, ry=0, rz=4, total=4
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# Gate count scales linearly with reps
for reps in [1, 2, 3, 5]:
enc_r = HigherOrderAngleEncoding(n_features=4, order=2, reps=reps)
bd = enc_r.gate_count_breakdown()
print(f"reps={reps}: total={bd['total']} (= {enc_r.n_qubits} qubits × {reps} reps)")
# Gate count scales linearly with reps
for reps in [1, 2, 3, 5]:
enc_r = HigherOrderAngleEncoding(n_features=4, order=2, reps=reps)
bd = enc_r.gate_count_breakdown()
print(f"reps={reps}: total={bd['total']} (= {enc_r.n_qubits} qubits × {reps} reps)")
reps=1: total=4 (= 4 qubits × 1 reps) reps=2: total=8 (= 4 qubits × 2 reps) reps=3: total=12 (= 4 qubits × 3 reps) reps=5: total=20 (= 4 qubits × 5 reps)
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# Two-qubit gates are ALWAYS zero (no entanglement)
enc_big = HigherOrderAngleEncoding(n_features=8, order=4)
bd = enc_big.gate_count_breakdown()
print(f"n_features=8, order=4: total_two_qubit = {bd['total_two_qubit']}")
assert bd["total_two_qubit"] == 0
# Two-qubit gates are ALWAYS zero (no entanglement)
enc_big = HigherOrderAngleEncoding(n_features=8, order=4)
bd = enc_big.gate_count_breakdown()
print(f"n_features=8, order=4: total_two_qubit = {bd['total_two_qubit']}")
assert bd["total_two_qubit"] == 0
C:\Users\ashut\AppData\Local\Temp\ipykernel_49516\399513000.py:2: UserWarning: High term count (162 terms) may impact performance. With order=4 and n_features=8, the term count grows combinatorially. Consider using a lower order for better performance, or ensure this complexity is intentional. enc_big = HigherOrderAngleEncoding(n_features=8, order=4) High term count: 162 terms with order=4, n_features=8
n_features=8, order=4: total_two_qubit = 0
11. Resource Summary¶
resource_summary() provides a comprehensive view of circuit resources, polynomial configuration, and hardware requirements.
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enc = HigherOrderAngleEncoding(n_features=4, order=2, rotation="Y", reps=2, scaling=1.5)
summary = enc.resource_summary()
print("=== Resource Summary ===")
print(f"\n--- Circuit Structure ---")
print(f" n_qubits: {summary['n_qubits']}")
print(f" n_features: {summary['n_features']}")
print(f" depth: {summary['depth']}")
print(f" reps: {summary['reps']}")
print(f" rotation: {summary['rotation']}")
print(f"\n--- Polynomial Configuration ---")
print(f" order: {summary['order']}")
print(f" combination: {summary['combination']}")
print(f" include_first_order: {summary['include_first_order']}")
print(f" scaling: {summary['scaling']}")
print(f" n_terms: {summary['n_terms']}")
print(f" terms_by_order: {summary['terms_by_order']}")
print(f"\n--- Gate Counts ---")
for k, v in summary["gate_counts"].items():
print(f" {k}: {v}")
print(f"\n--- Encoding Characteristics ---")
print(f" is_entangling: {summary['is_entangling']}")
print(f" simulability: {summary['simulability']}")
print(f" trainability_estimate: {summary['trainability_estimate']:.4f}")
print(f"\n--- Hardware Requirements ---")
print(f" connectivity: {summary['hardware_requirements']['connectivity']}")
print(f" native_gates: {summary['hardware_requirements']['native_gates']}")
enc = HigherOrderAngleEncoding(n_features=4, order=2, rotation="Y", reps=2, scaling=1.5)
summary = enc.resource_summary()
print("=== Resource Summary ===")
print(f"\n--- Circuit Structure ---")
print(f" n_qubits: {summary['n_qubits']}")
print(f" n_features: {summary['n_features']}")
print(f" depth: {summary['depth']}")
print(f" reps: {summary['reps']}")
print(f" rotation: {summary['rotation']}")
print(f"\n--- Polynomial Configuration ---")
print(f" order: {summary['order']}")
print(f" combination: {summary['combination']}")
print(f" include_first_order: {summary['include_first_order']}")
print(f" scaling: {summary['scaling']}")
print(f" n_terms: {summary['n_terms']}")
print(f" terms_by_order: {summary['terms_by_order']}")
print(f"\n--- Gate Counts ---")
for k, v in summary["gate_counts"].items():
print(f" {k}: {v}")
print(f"\n--- Encoding Characteristics ---")
print(f" is_entangling: {summary['is_entangling']}")
print(f" simulability: {summary['simulability']}")
print(f" trainability_estimate: {summary['trainability_estimate']:.4f}")
print(f"\n--- Hardware Requirements ---")
print(f" connectivity: {summary['hardware_requirements']['connectivity']}")
print(f" native_gates: {summary['hardware_requirements']['native_gates']}")
=== Resource Summary ===
--- Circuit Structure ---
n_qubits: 4
n_features: 4
depth: 2
reps: 2
rotation: Y
--- Polynomial Configuration ---
order: 2
combination: product
include_first_order: True
scaling: 1.5
n_terms: 10
terms_by_order: {1: 4, 2: 6}
--- Gate Counts ---
rx: 0
ry: 8
rz: 0
total_single_qubit: 8
total_two_qubit: 0
total: 8
--- Encoding Characteristics ---
is_entangling: False
simulability: simulable
trainability_estimate: 0.9100
--- Hardware Requirements ---
connectivity: none
native_gates: ['RY']
12. EncodingProperties (Base Class)¶
The properties attribute returns a frozen EncodingProperties dataclass computed from the encoding configuration. It is lazily computed and cached (thread-safe).
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from encoding_atlas.core.properties import EncodingProperties
enc = HigherOrderAngleEncoding(n_features=4, order=2, reps=2)
props = enc.properties
print(f"Type: {type(props).__name__}")
print(f"Frozen (immutable): {props.__class__.__dataclass_params__.frozen}")
print()
print(f"n_qubits: {props.n_qubits}")
print(f"depth: {props.depth}")
print(f"gate_count: {props.gate_count}")
print(f"single_qubit_gates: {props.single_qubit_gates}")
print(f"two_qubit_gates: {props.two_qubit_gates}")
print(f"parameter_count: {props.parameter_count} (all data-dependent, no trainable params)")
print(f"is_entangling: {props.is_entangling}")
print(f"simulability: {props.simulability}")
print(f"trainability_estimate: {props.trainability_estimate}")
print(f"notes: {props.notes}")
from encoding_atlas.core.properties import EncodingProperties
enc = HigherOrderAngleEncoding(n_features=4, order=2, reps=2)
props = enc.properties
print(f"Type: {type(props).__name__}")
print(f"Frozen (immutable): {props.__class__.__dataclass_params__.frozen}")
print()
print(f"n_qubits: {props.n_qubits}")
print(f"depth: {props.depth}")
print(f"gate_count: {props.gate_count}")
print(f"single_qubit_gates: {props.single_qubit_gates}")
print(f"two_qubit_gates: {props.two_qubit_gates}")
print(f"parameter_count: {props.parameter_count} (all data-dependent, no trainable params)")
print(f"is_entangling: {props.is_entangling}")
print(f"simulability: {props.simulability}")
print(f"trainability_estimate: {props.trainability_estimate}")
print(f"notes: {props.notes}")
Type: EncodingProperties Frozen (immutable): True n_qubits: 4 depth: 2 gate_count: 8 single_qubit_gates: 8 two_qubit_gates: 0 parameter_count: 0 (all data-dependent, no trainable params) is_entangling: False simulability: simulable trainability_estimate: 0.9099999999999999 notes: Order-2 polynomial encoding, with 10 terms, (product combination)
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# Properties are cached — same object returned
props1 = enc.properties
props2 = enc.properties
print(f"Same object: {props1 is props2}")
# Properties are cached — same object returned
props1 = enc.properties
props2 = enc.properties
print(f"Same object: {props1 is props2}")
Same object: True
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# The to_dict() method converts to a plain dictionary
props_dict = props.to_dict()
print("Properties as dict:")
for k, v in props_dict.items():
print(f" {k}: {v}")
# The to_dict() method converts to a plain dictionary
props_dict = props.to_dict()
print("Properties as dict:")
for k, v in props_dict.items():
print(f" {k}: {v}")
Properties as dict: n_qubits: 4 depth: 2 gate_count: 8 single_qubit_gates: 8 two_qubit_gates: 0 parameter_count: 0 is_entangling: False simulability: simulable expressibility: None entanglement_capability: None trainability_estimate: 0.9099999999999999 noise_resilience_estimate: None notes: Order-2 polynomial encoding, with 10 terms, (product combination)
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# Trying to modify a frozen property raises an error
try:
props.n_qubits = 999
except AttributeError as e:
print(f"Cannot modify frozen dataclass: {e}")
# Trying to modify a frozen property raises an error
try:
props.n_qubits = 999
except AttributeError as e:
print(f"Cannot modify frozen dataclass: {e}")
Cannot modify frozen dataclass: cannot assign to field 'n_qubits'
13. Capability Protocols¶
The library uses structural subtyping protocols (PEP 544) to define optional capabilities. You can check at runtime what an encoding supports.
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from encoding_atlas.core.protocols import (
ResourceAnalyzable,
DataDependentResourceAnalyzable,
EntanglementQueryable,
DataTransformable,
is_resource_analyzable,
is_entanglement_queryable,
is_data_transformable,
)
enc = HigherOrderAngleEncoding(n_features=4, order=2)
print("Protocol checks for HigherOrderAngleEncoding:")
print(f" ResourceAnalyzable: {isinstance(enc, ResourceAnalyzable)}")
print(f" DataDependentResourceAnalyzable: {isinstance(enc, DataDependentResourceAnalyzable)}")
print(f" EntanglementQueryable: {isinstance(enc, EntanglementQueryable)}")
print(f" DataTransformable: {isinstance(enc, DataTransformable)}")
# Convenience functions
print(f"\nConvenience functions:")
print(f" is_resource_analyzable: {is_resource_analyzable(enc)}")
print(f" is_entanglement_queryable: {is_entanglement_queryable(enc)}")
print(f" is_data_transformable: {is_data_transformable(enc)}")
from encoding_atlas.core.protocols import (
ResourceAnalyzable,
DataDependentResourceAnalyzable,
EntanglementQueryable,
DataTransformable,
is_resource_analyzable,
is_entanglement_queryable,
is_data_transformable,
)
enc = HigherOrderAngleEncoding(n_features=4, order=2)
print("Protocol checks for HigherOrderAngleEncoding:")
print(f" ResourceAnalyzable: {isinstance(enc, ResourceAnalyzable)}")
print(f" DataDependentResourceAnalyzable: {isinstance(enc, DataDependentResourceAnalyzable)}")
print(f" EntanglementQueryable: {isinstance(enc, EntanglementQueryable)}")
print(f" DataTransformable: {isinstance(enc, DataTransformable)}")
# Convenience functions
print(f"\nConvenience functions:")
print(f" is_resource_analyzable: {is_resource_analyzable(enc)}")
print(f" is_entanglement_queryable: {is_entanglement_queryable(enc)}")
print(f" is_data_transformable: {is_data_transformable(enc)}")
Protocol checks for HigherOrderAngleEncoding: ResourceAnalyzable: True DataDependentResourceAnalyzable: False EntanglementQueryable: False DataTransformable: False Convenience functions: is_resource_analyzable: True is_entanglement_queryable: False is_data_transformable: False
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# Generic function that works with any ResourceAnalyzable encoding
def analyze_encoding(enc):
if isinstance(enc, ResourceAnalyzable):
summary = enc.resource_summary()
print(f"{enc.__class__.__name__}:")
print(f" Gates: {summary['gate_counts']['total']}")
print(f" Entangling: {summary['is_entangling']}")
print(f" Simulable: {summary['simulability']}")
else:
print(f"{enc.__class__.__name__}: Resource analysis not supported")
analyze_encoding(HigherOrderAngleEncoding(n_features=4, order=2))
# Generic function that works with any ResourceAnalyzable encoding
def analyze_encoding(enc):
if isinstance(enc, ResourceAnalyzable):
summary = enc.resource_summary()
print(f"{enc.__class__.__name__}:")
print(f" Gates: {summary['gate_counts']['total']}")
print(f" Entangling: {summary['is_entangling']}")
print(f" Simulable: {summary['simulability']}")
else:
print(f"{enc.__class__.__name__}: Resource analysis not supported")
analyze_encoding(HigherOrderAngleEncoding(n_features=4, order=2))
HigherOrderAngleEncoding: Gates: 4 Entangling: False Simulable: simulable
14. Registry & Factory Access¶
Encodings are registered under string names for factory-style creation.
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from encoding_atlas import get_encoding, list_encodings
# List all registered encoding names
all_names = list_encodings()
print("Registered encodings:")
for name in all_names:
print(f" - {name}")
from encoding_atlas import get_encoding, list_encodings
# List all registered encoding names
all_names = list_encodings()
print("Registered encodings:")
for name in all_names:
print(f" - {name}")
Registered encodings: - amplitude - angle - angle_ry - basis - covariant - covariant_feature_map - cyclic_equivariant - cyclic_equivariant_feature_map - data_reuploading - hamiltonian - hamiltonian_encoding - hardware_efficient - higher_order_angle - iqp - pauli_feature_map - qaoa - qaoa_encoding - so2_equivariant - so2_equivariant_feature_map - swap_equivariant - swap_equivariant_feature_map - symmetry_inspired - symmetry_inspired_feature_map - trainable - trainable_encoding - zz_feature_map
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# Create HigherOrderAngleEncoding via the registry
enc_from_registry = get_encoding("higher_order_angle", n_features=4, order=2)
print(f"Type: {type(enc_from_registry).__name__}")
print(f"n_features: {enc_from_registry.n_features}")
print(f"order: {enc_from_registry.order}")
# Create HigherOrderAngleEncoding via the registry
enc_from_registry = get_encoding("higher_order_angle", n_features=4, order=2)
print(f"Type: {type(enc_from_registry).__name__}")
print(f"n_features: {enc_from_registry.n_features}")
print(f"order: {enc_from_registry.order}")
Type: HigherOrderAngleEncoding n_features: 4 order: 2
15. Encoding Guide & Recommendations¶
The library includes a recommendation system to help choose encodings.
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from encoding_atlas.guide import recommend_encoding, Recommendation
rec = recommend_encoding(n_features=4, n_samples=500, priority="accuracy")
print(f"Recommended: {rec.encoding_name}")
print(f"Explanation: {rec.explanation}")
print(f"Alternatives: {rec.alternatives}")
print(f"Confidence: {rec.confidence}")
from encoding_atlas.guide import recommend_encoding, Recommendation
rec = recommend_encoding(n_features=4, n_samples=500, priority="accuracy")
print(f"Recommended: {rec.encoding_name}")
print(f"Explanation: {rec.explanation}")
print(f"Alternatives: {rec.alternatives}")
print(f"Confidence: {rec.confidence}")
Recommended: iqp Explanation: IQP encoding creates highly entangled states with provable classical simulation hardness, well-suited for kernel methods Alternatives: ['data_reuploading', 'zz_feature_map', 'pauli_feature_map'] Confidence: 0.74
16. Equality, Hashing & Collections¶
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enc1 = HigherOrderAngleEncoding(n_features=4, order=2)
enc2 = HigherOrderAngleEncoding(n_features=4, order=2)
enc3 = HigherOrderAngleEncoding(n_features=4, order=3)
print(f"enc1 == enc2 (same params): {enc1 == enc2}")
print(f"enc1 == enc3 (diff order): {enc1 == enc3}")
print(f"hash(enc1) == hash(enc2): {hash(enc1) == hash(enc2)}")
# Works in sets and as dict keys
s = {enc1, enc2, enc3}
print(f"\nSet {{enc1, enc2, enc3}} has {len(s)} unique elements (enc1==enc2)")
d = {enc1: "result_A"}
print(f"Dict lookup d[enc2] = {d[enc2]!r} (enc2 finds enc1's entry)")
enc1 = HigherOrderAngleEncoding(n_features=4, order=2)
enc2 = HigherOrderAngleEncoding(n_features=4, order=2)
enc3 = HigherOrderAngleEncoding(n_features=4, order=3)
print(f"enc1 == enc2 (same params): {enc1 == enc2}")
print(f"enc1 == enc3 (diff order): {enc1 == enc3}")
print(f"hash(enc1) == hash(enc2): {hash(enc1) == hash(enc2)}")
# Works in sets and as dict keys
s = {enc1, enc2, enc3}
print(f"\nSet {{enc1, enc2, enc3}} has {len(s)} unique elements (enc1==enc2)")
d = {enc1: "result_A"}
print(f"Dict lookup d[enc2] = {d[enc2]!r} (enc2 finds enc1's entry)")
enc1 == enc2 (same params): True
enc1 == enc3 (diff order): False
hash(enc1) == hash(enc2): True
Set {enc1, enc2, enc3} has 2 unique elements (enc1==enc2)
Dict lookup d[enc2] = 'result_A' (enc2 finds enc1's entry)
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# Inequality across all parameter variations
params = [
("n_features", HigherOrderAngleEncoding(n_features=8, order=2)),
("order", HigherOrderAngleEncoding(n_features=4, order=3)),
("rotation", HigherOrderAngleEncoding(n_features=4, rotation="Z")),
("combination", HigherOrderAngleEncoding(n_features=4, combination="sum")),
("scaling", HigherOrderAngleEncoding(n_features=4, scaling=2.0)),
("reps", HigherOrderAngleEncoding(n_features=4, reps=3)),
]
base = HigherOrderAngleEncoding(n_features=4, order=2)
for name, other in params:
print(f"Different {name:20s}: equal={base == other}")
# Inequality across all parameter variations
params = [
("n_features", HigherOrderAngleEncoding(n_features=8, order=2)),
("order", HigherOrderAngleEncoding(n_features=4, order=3)),
("rotation", HigherOrderAngleEncoding(n_features=4, rotation="Z")),
("combination", HigherOrderAngleEncoding(n_features=4, combination="sum")),
("scaling", HigherOrderAngleEncoding(n_features=4, scaling=2.0)),
("reps", HigherOrderAngleEncoding(n_features=4, reps=3)),
]
base = HigherOrderAngleEncoding(n_features=4, order=2)
for name, other in params:
print(f"Different {name:20s}: equal={base == other}")
Different n_features : equal=False Different order : equal=False Different rotation : equal=False Different combination : equal=False Different scaling : equal=False Different reps : equal=False
17. String Representation¶
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enc = HigherOrderAngleEncoding(
n_features=5, order=3, rotation="Z", combination="sum",
include_first_order=False, scaling=2.5, reps=2,
)
r = repr(enc)
print(f"repr: {r}")
print(f"\nContains all parameters:")
for param in ["n_features=5", "order=3", "rotation='Z'", "combination='sum'",
"include_first_order=False", "scaling=2.5", "reps=2"]:
print(f" {param:30s} {'✓' if param in r else '✗'}")
enc = HigherOrderAngleEncoding(
n_features=5, order=3, rotation="Z", combination="sum",
include_first_order=False, scaling=2.5, reps=2,
)
r = repr(enc)
print(f"repr: {r}")
print(f"\nContains all parameters:")
for param in ["n_features=5", "order=3", "rotation='Z'", "combination='sum'",
"include_first_order=False", "scaling=2.5", "reps=2"]:
print(f" {param:30s} {'✓' if param in r else '✗'}")
repr: HigherOrderAngleEncoding(n_features=5, order=3, rotation='Z', combination='sum', include_first_order=False, scaling=2.5, reps=2) Contains all parameters: n_features=5 ✓ order=3 ✓ rotation='Z' ✓ combination='sum' ✓ include_first_order=False ✓ scaling=2.5 ✓ reps=2 ✓
18. Serialization (Pickle)¶
Encodings support full pickle serialization, including cached properties.
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enc = HigherOrderAngleEncoding(
n_features=4, order=3, rotation="Z", combination="sum", scaling=2.0, reps=2,
)
# Force properties to be computed before pickling
_ = enc.properties
# Pickle roundtrip
pickled = pickle.dumps(enc)
restored = pickle.loads(pickled)
print(f"Original: {enc}")
print(f"Restored: {restored}")
print(f"Equal: {enc == restored}")
print(f"Hash match:{hash(enc) == hash(restored)}")
enc = HigherOrderAngleEncoding(
n_features=4, order=3, rotation="Z", combination="sum", scaling=2.0, reps=2,
)
# Force properties to be computed before pickling
_ = enc.properties
# Pickle roundtrip
pickled = pickle.dumps(enc)
restored = pickle.loads(pickled)
print(f"Original: {enc}")
print(f"Restored: {restored}")
print(f"Equal: {enc == restored}")
print(f"Hash match:{hash(enc) == hash(restored)}")
Original: HigherOrderAngleEncoding(n_features=4, order=3, rotation='Z', combination='sum', include_first_order=True, scaling=2.0, reps=2) Restored: HigherOrderAngleEncoding(n_features=4, order=3, rotation='Z', combination='sum', include_first_order=True, scaling=2.0, reps=2) Equal: True Hash match:True
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# Circuit generation works after unpickling
x = np.array([0.1, 0.2, 0.3, 0.4])
circuit = restored.get_circuit(x, backend="pennylane")
print(f"Circuit callable after restore: {callable(circuit)}")
# Properties work after unpickling
props = restored.properties
print(f"Properties accessible: gate_count={props.gate_count}")
# Circuit generation works after unpickling
x = np.array([0.1, 0.2, 0.3, 0.4])
circuit = restored.get_circuit(x, backend="pennylane")
print(f"Circuit callable after restore: {callable(circuit)}")
# Properties work after unpickling
props = restored.properties
print(f"Properties accessible: gate_count={props.gate_count}")
Circuit callable after restore: True Properties accessible: gate_count=8
19. Thread Safety & Concurrent Access¶
The encoding is immutable after construction — all read operations (including circuit generation) are thread-safe.
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from concurrent.futures import ThreadPoolExecutor, as_completed
enc = HigherOrderAngleEncoding(n_features=4, order=2)
num_threads = 8
circuits_per_thread = 20
errors = []
def generate_circuits(thread_id):
results = []
try:
for i in range(circuits_per_thread):
rng = np.random.default_rng(seed=thread_id * 1000 + i)
x = rng.standard_normal(4)
circuit = enc.get_circuit(x, backend="pennylane")
results.append(circuit)
except Exception as e:
errors.append(e)
return results
with ThreadPoolExecutor(max_workers=num_threads) as executor:
futures = [executor.submit(generate_circuits, i) for i in range(num_threads)]
all_results = [f.result() for f in as_completed(futures)]
total = sum(len(r) for r in all_results)
print(f"Threads: {num_threads}, Circuits per thread: {circuits_per_thread}")
print(f"Total circuits generated: {total}")
print(f"Errors: {len(errors)}")
assert len(errors) == 0 and total == num_threads * circuits_per_thread
from concurrent.futures import ThreadPoolExecutor, as_completed
enc = HigherOrderAngleEncoding(n_features=4, order=2)
num_threads = 8
circuits_per_thread = 20
errors = []
def generate_circuits(thread_id):
results = []
try:
for i in range(circuits_per_thread):
rng = np.random.default_rng(seed=thread_id * 1000 + i)
x = rng.standard_normal(4)
circuit = enc.get_circuit(x, backend="pennylane")
results.append(circuit)
except Exception as e:
errors.append(e)
return results
with ThreadPoolExecutor(max_workers=num_threads) as executor:
futures = [executor.submit(generate_circuits, i) for i in range(num_threads)]
all_results = [f.result() for f in as_completed(futures)]
total = sum(len(r) for r in all_results)
print(f"Threads: {num_threads}, Circuits per thread: {circuits_per_thread}")
print(f"Total circuits generated: {total}")
print(f"Errors: {len(errors)}")
assert len(errors) == 0 and total == num_threads * circuits_per_thread
Threads: 8, Circuits per thread: 20 Total circuits generated: 160 Errors: 0
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# Concurrent property access — all return the same cached object
results = []
def access_properties():
results.append(enc.properties)
with ThreadPoolExecutor(max_workers=20) as executor:
futures = [executor.submit(access_properties) for _ in range(20)]
for f in as_completed(futures):
f.result()
print(f"All {len(results)} accesses returned same object: {all(r is results[0] for r in results)}")
# Concurrent property access — all return the same cached object
results = []
def access_properties():
results.append(enc.properties)
with ThreadPoolExecutor(max_workers=20) as executor:
futures = [executor.submit(access_properties) for _ in range(20)]
for f in as_completed(futures):
f.result()
print(f"All {len(results)} accesses returned same object: {all(r is results[0] for r in results)}")
All 20 accesses returned same object: True
20. Edge Cases & Numerical Stability¶
20.1 Single Feature (Minimum Configuration)¶
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enc_min = HigherOrderAngleEncoding(n_features=1, order=1)
x = np.array([0.5])
print(f"n_features=1: n_qubits={enc_min.n_qubits}, n_terms={enc_min.n_terms}")
circuit = enc_min.get_circuit(x, backend="pennylane")
print(f"Circuit callable: {callable(circuit)}")
enc_min = HigherOrderAngleEncoding(n_features=1, order=1)
x = np.array([0.5])
print(f"n_features=1: n_qubits={enc_min.n_qubits}, n_terms={enc_min.n_terms}")
circuit = enc_min.get_circuit(x, backend="pennylane")
print(f"Circuit callable: {callable(circuit)}")
n_features=1: n_qubits=1, n_terms=1 Circuit callable: True
20.2 Zero Input¶
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enc = HigherOrderAngleEncoding(n_features=4, order=2, combination="product")
x_zero = np.zeros(4)
angles = enc.compute_angles(x_zero)
print(f"Zero input angles: {angles}")
assert np.allclose(angles, 0.0), "All angles should be zero"
# Zero input still produces a valid quantum state (|0...0⟩)
dev = qml.device("default.qubit", wires=4)
fn = enc.get_circuit(x_zero, backend="pennylane")
@qml.qnode(dev)
def zero_circuit():
fn()
return qml.state()
state = zero_circuit()
print(f"State norm: {np.sum(np.abs(state)**2):.10f}")
print(f"|0000⟩ amplitude: {np.abs(state[0]):.10f}")
enc = HigherOrderAngleEncoding(n_features=4, order=2, combination="product")
x_zero = np.zeros(4)
angles = enc.compute_angles(x_zero)
print(f"Zero input angles: {angles}")
assert np.allclose(angles, 0.0), "All angles should be zero"
# Zero input still produces a valid quantum state (|0...0⟩)
dev = qml.device("default.qubit", wires=4)
fn = enc.get_circuit(x_zero, backend="pennylane")
@qml.qnode(dev)
def zero_circuit():
fn()
return qml.state()
state = zero_circuit()
print(f"State norm: {np.sum(np.abs(state)**2):.10f}")
print(f"|0000⟩ amplitude: {np.abs(state[0]):.10f}")
Zero input angles: [0. 0. 0. 0.] State norm: 1.0000000000 |0000⟩ amplitude: 1.0000000000
20.3 Negative Values & Mixed Signs¶
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enc = HigherOrderAngleEncoding(n_features=4, order=2, combination="product")
x_neg = np.array([-1.0, -2.0, -3.0, -4.0])
x_mixed = np.array([1.0, -1.0, 1.0, -1.0])
angles_neg = enc.compute_angles(x_neg)
angles_mixed = enc.compute_angles(x_mixed)
print(f"All negative: angles = {angles_neg}")
print(f"Mixed signs: angles = {angles_mixed}")
# Both produce valid circuits
for label, x in [("negative", x_neg), ("mixed", x_mixed)]:
fn = enc.get_circuit(x, backend="pennylane")
dev = qml.device("default.qubit", wires=4)
@qml.qnode(dev)
def check():
fn()
return qml.state()
state = check()
print(f" {label}: norm = {np.sum(np.abs(state)**2):.10f}")
enc = HigherOrderAngleEncoding(n_features=4, order=2, combination="product")
x_neg = np.array([-1.0, -2.0, -3.0, -4.0])
x_mixed = np.array([1.0, -1.0, 1.0, -1.0])
angles_neg = enc.compute_angles(x_neg)
angles_mixed = enc.compute_angles(x_mixed)
print(f"All negative: angles = {angles_neg}")
print(f"Mixed signs: angles = {angles_mixed}")
# Both produce valid circuits
for label, x in [("negative", x_neg), ("mixed", x_mixed)]:
fn = enc.get_circuit(x, backend="pennylane")
dev = qml.device("default.qubit", wires=4)
@qml.qnode(dev)
def check():
fn()
return qml.state()
state = check()
print(f" {label}: norm = {np.sum(np.abs(state)**2):.10f}")
All negative: angles = [ 9. 13. 1. 2.] Mixed signs: angles = [ 1. -1. 0. -2.] negative: norm = 1.0000000000 mixed: norm = 1.0000000000
20.4 Extreme Values¶
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enc = HigherOrderAngleEncoding(n_features=4, order=2)
dev = qml.device("default.qubit", wires=4)
test_cases = {
"very small": np.array([1e-15, 1e-16, 1e-17, 1e-18]),
"very large": np.array([1e5, 2e5, 3e5, 4e5]),
"mixed extremes": np.array([1e-10, 1e10, 1e-5, 1e5]),
"near pi": np.array([np.pi - 1e-14, np.pi + 1e-14, 2*np.pi, np.pi/2]),
"machine epsilon": np.finfo(float).eps * np.array([1, 2, 3, 4]),
}
for label, x in test_cases.items():
fn = enc.get_circuit(x, backend="pennylane")
@qml.qnode(dev)
def test_circuit():
fn()
return qml.state()
state = test_circuit()
norm = np.sum(np.abs(state)**2)
finite = np.all(np.isfinite(state))
print(f" {label:17s}: norm={norm:.10f}, all_finite={finite}")
enc = HigherOrderAngleEncoding(n_features=4, order=2)
dev = qml.device("default.qubit", wires=4)
test_cases = {
"very small": np.array([1e-15, 1e-16, 1e-17, 1e-18]),
"very large": np.array([1e5, 2e5, 3e5, 4e5]),
"mixed extremes": np.array([1e-10, 1e10, 1e-5, 1e5]),
"near pi": np.array([np.pi - 1e-14, np.pi + 1e-14, 2*np.pi, np.pi/2]),
"machine epsilon": np.finfo(float).eps * np.array([1, 2, 3, 4]),
}
for label, x in test_cases.items():
fn = enc.get_circuit(x, backend="pennylane")
@qml.qnode(dev)
def test_circuit():
fn()
return qml.state()
state = test_circuit()
norm = np.sum(np.abs(state)**2)
finite = np.all(np.isfinite(state))
print(f" {label:17s}: norm={norm:.10f}, all_finite={finite}")
very small : norm=1.0000000000, all_finite=True very large : norm=1.0000000000, all_finite=True mixed extremes : norm=1.0000000000, all_finite=True near pi : norm=1.0000000000, all_finite=True machine epsilon : norm=1.0000000000, all_finite=True
20.5 Negative Scaling Reverses Angles¶
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enc_pos = HigherOrderAngleEncoding(n_features=3, order=1, scaling=1.0)
enc_neg = HigherOrderAngleEncoding(n_features=3, order=1, scaling=-1.0)
x = np.array([0.1, 0.2, 0.3])
angles_pos = enc_pos.compute_angles(x)
angles_neg = enc_neg.compute_angles(x)
print(f"scaling=+1.0: {angles_pos}")
print(f"scaling=-1.0: {angles_neg}")
print(f"Sum to zero: {np.allclose(angles_pos + angles_neg, 0.0)}")
enc_pos = HigherOrderAngleEncoding(n_features=3, order=1, scaling=1.0)
enc_neg = HigherOrderAngleEncoding(n_features=3, order=1, scaling=-1.0)
x = np.array([0.1, 0.2, 0.3])
angles_pos = enc_pos.compute_angles(x)
angles_neg = enc_neg.compute_angles(x)
print(f"scaling=+1.0: {angles_pos}")
print(f"scaling=-1.0: {angles_neg}")
print(f"Sum to zero: {np.allclose(angles_pos + angles_neg, 0.0)}")
scaling=+1.0: [0.1 0.2 0.3] scaling=-1.0: [-0.1 -0.2 -0.3] Sum to zero: True
20.6 Zero Scaling¶
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enc_zero = HigherOrderAngleEncoding(n_features=4, order=2, scaling=0.0)
x = np.array([100.0, 200.0, 300.0, 400.0])
angles = enc_zero.compute_angles(x)
print(f"Zero scaling, large input: angles = {angles}")
assert np.allclose(angles, 0.0)
enc_zero = HigherOrderAngleEncoding(n_features=4, order=2, scaling=0.0)
x = np.array([100.0, 200.0, 300.0, 400.0])
angles = enc_zero.compute_angles(x)
print(f"Zero scaling, large input: angles = {angles}")
assert np.allclose(angles, 0.0)
Zero scaling, large input: angles = [0. 0. 0. 0.]
20.7 High Term Count Warning¶
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# When term count exceeds 100, a UserWarning is issued
with warnings.catch_warnings(record=True) as w:
warnings.simplefilter("always")
enc_warn = HigherOrderAngleEncoding(n_features=10, order=5)
if w:
print(f"Warning issued: {w[0].category.__name__}")
print(f"Message: {w[0].message}")
else:
print(f"No warning (term count = {enc_warn.n_terms})")
# Verify threshold
print(f"\nTerm count for n=10, order=5: {enc_warn.n_terms}")
# When term count exceeds 100, a UserWarning is issued
with warnings.catch_warnings(record=True) as w:
warnings.simplefilter("always")
enc_warn = HigherOrderAngleEncoding(n_features=10, order=5)
if w:
print(f"Warning issued: {w[0].category.__name__}")
print(f"Message: {w[0].message}")
else:
print(f"No warning (term count = {enc_warn.n_terms})")
# Verify threshold
print(f"\nTerm count for n=10, order=5: {enc_warn.n_terms}")
High term count: 637 terms with order=5, n_features=10
Warning issued: UserWarning Message: High term count (637 terms) may impact performance. With order=5 and n_features=10, the term count grows combinatorially. Consider using a lower order for better performance, or ensure this complexity is intentional. Term count for n=10, order=5: 637
21. Comparison: product vs sum Combination¶
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x = np.array([2.0, 3.0, 5.0])
enc_prod = HigherOrderAngleEncoding(n_features=3, order=2, combination="product")
enc_sum = HigherOrderAngleEncoding(n_features=3, order=2, combination="sum")
print("Terms and their values for x = [2, 3, 5]:")
print(f"{'Term':12s} {'Product':>10s} {'Sum':>10s}")
print("-" * 35)
for term in enc_prod.terms:
prod_val = np.prod([x[i] for i in term])
sum_val = np.sum([x[i] for i in term])
label = " × ".join(f"x_{j}" for j in term)
print(f"{label:12s} {prod_val:10.1f} {sum_val:10.1f}")
print(f"\nAngles (product): {enc_prod.compute_angles(x)}")
print(f"Angles (sum): {enc_sum.compute_angles(x)}")
x = np.array([2.0, 3.0, 5.0])
enc_prod = HigherOrderAngleEncoding(n_features=3, order=2, combination="product")
enc_sum = HigherOrderAngleEncoding(n_features=3, order=2, combination="sum")
print("Terms and their values for x = [2, 3, 5]:")
print(f"{'Term':12s} {'Product':>10s} {'Sum':>10s}")
print("-" * 35)
for term in enc_prod.terms:
prod_val = np.prod([x[i] for i in term])
sum_val = np.sum([x[i] for i in term])
label = " × ".join(f"x_{j}" for j in term)
print(f"{label:12s} {prod_val:10.1f} {sum_val:10.1f}")
print(f"\nAngles (product): {enc_prod.compute_angles(x)}")
print(f"Angles (sum): {enc_sum.compute_angles(x)}")
Terms and their values for x = [2, 3, 5]: Term Product Sum ----------------------------------- x_0 2.0 2.0 x_1 3.0 3.0 x_2 5.0 5.0 x_0 × x_1 6.0 5.0 x_0 × x_2 10.0 7.0 x_1 × x_2 15.0 8.0 Angles (product): [ 8. 13. 20.] Angles (sum): [ 7. 10. 13.]
22. Comparison: With vs Without First-Order Terms¶
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enc_with = HigherOrderAngleEncoding(n_features=4, order=2, include_first_order=True)
enc_without = HigherOrderAngleEncoding(n_features=4, order=2, include_first_order=False)
print(f"With first-order: {enc_with.n_terms} terms")
print(f"Without first-order: {enc_without.n_terms} terms")
print()
x = np.array([1.0, 2.0, 3.0, 4.0])
angles_with = enc_with.compute_angles(x)
angles_without = enc_without.compute_angles(x)
print(f"Angles with: {angles_with}")
print(f"Angles without: {angles_without}")
print()
print("Terms with first-order:")
for t in enc_with.terms:
print(f" {t}")
print("\nTerms without first-order:")
for t in enc_without.terms:
print(f" {t}")
enc_with = HigherOrderAngleEncoding(n_features=4, order=2, include_first_order=True)
enc_without = HigherOrderAngleEncoding(n_features=4, order=2, include_first_order=False)
print(f"With first-order: {enc_with.n_terms} terms")
print(f"Without first-order: {enc_without.n_terms} terms")
print()
x = np.array([1.0, 2.0, 3.0, 4.0])
angles_with = enc_with.compute_angles(x)
angles_without = enc_without.compute_angles(x)
print(f"Angles with: {angles_with}")
print(f"Angles without: {angles_without}")
print()
print("Terms with first-order:")
for t in enc_with.terms:
print(f" {t}")
print("\nTerms without first-order:")
for t in enc_without.terms:
print(f" {t}")
With first-order: 10 terms Without first-order: 6 terms Angles with: [11. 17. 7. 10.] Angles without: [10. 15. 4. 6.] Terms with first-order: (0,) (1,) (2,) (3,) (0, 1) (0, 2) (0, 3) (1, 2) (1, 3) (2, 3) Terms without first-order: (0, 1) (0, 2) (0, 3) (1, 2) (1, 3) (2, 3)
23. Scaling Behaviour¶
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enc_base = HigherOrderAngleEncoding(n_features=4, order=2, scaling=1.0)
x = np.array([0.5, 1.0, 1.5, 2.0])
print(f"{'Scaling':>10s} {'Angles':>40s}")
print("-" * 55)
for s in [0.1, 0.5, 1.0, 2.0, 5.0, 10.0]:
enc_s = HigherOrderAngleEncoding(n_features=4, order=2, scaling=s)
angles = enc_s.compute_angles(x)
print(f"{s:10.1f} {np.array2string(angles, precision=3, floatmode='fixed')}")
enc_base = HigherOrderAngleEncoding(n_features=4, order=2, scaling=1.0)
x = np.array([0.5, 1.0, 1.5, 2.0])
print(f"{'Scaling':>10s} {'Angles':>40s}")
print("-" * 55)
for s in [0.1, 0.5, 1.0, 2.0, 5.0, 10.0]:
enc_s = HigherOrderAngleEncoding(n_features=4, order=2, scaling=s)
angles = enc_s.compute_angles(x)
print(f"{s:10.1f} {np.array2string(angles, precision=3, floatmode='fixed')}")
Scaling Angles
-------------------------------------------------------
0.1 [0.300 0.475 0.250 0.350]
0.5 [1.500 2.375 1.250 1.750]
1.0 [3.000 4.750 2.500 3.500]
2.0 [6.000 9.500 5.000 7.000]
5.0 [15.000 23.750 12.500 17.500]
10.0 [30.000 47.500 25.000 35.000]
24. Term Count Utility Function¶
The count_terms() function computes the number of terms without creating an encoding.
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from encoding_atlas.encodings.higher_order_angle import count_terms
# Verify formula: sum of C(n, k) for k = start..order
print(f"{'n':>3s} {'order':>6s} {'1st':>5s} {'count':>6s} {'formula':>8s}")
print("-" * 35)
for n in [3, 4, 5, 8, 10]:
for order in [1, 2, 3, min(n, 4)]:
for incl in [True, False]:
if order == 1 and not incl:
continue
c = count_terms(n, order, incl)
start = 1 if incl else 2
expected = sum(math.comb(n, k) for k in range(start, order + 1))
assert c == expected
if incl:
print(f"{n:3d} {order:6d} {'yes':>5s} {c:6d} {expected:8d}")
from encoding_atlas.encodings.higher_order_angle import count_terms
# Verify formula: sum of C(n, k) for k = start..order
print(f"{'n':>3s} {'order':>6s} {'1st':>5s} {'count':>6s} {'formula':>8s}")
print("-" * 35)
for n in [3, 4, 5, 8, 10]:
for order in [1, 2, 3, min(n, 4)]:
for incl in [True, False]:
if order == 1 and not incl:
continue
c = count_terms(n, order, incl)
start = 1 if incl else 2
expected = sum(math.comb(n, k) for k in range(start, order + 1))
assert c == expected
if incl:
print(f"{n:3d} {order:6d} {'yes':>5s} {c:6d} {expected:8d}")
n order 1st count formula ----------------------------------- 3 1 yes 3 3 3 2 yes 6 6 3 3 yes 7 7 3 3 yes 7 7 4 1 yes 4 4 4 2 yes 10 10 4 3 yes 14 14 4 4 yes 15 15 5 1 yes 5 5 5 2 yes 15 15 5 3 yes 25 25 5 4 yes 30 30 8 1 yes 8 8 8 2 yes 36 36 8 3 yes 92 92 8 4 yes 162 162 10 1 yes 10 10 10 2 yes 55 55 10 3 yes 175 175 10 4 yes 385 385
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# count_terms matches actual encoding term count
for n in range(2, 7):
for order in range(1, n + 1):
for incl in [True, False]:
if order == 1 and not incl:
continue
expected = count_terms(n, order, incl)
enc = HigherOrderAngleEncoding(n_features=n, order=order, include_first_order=incl)
assert enc.n_terms == expected, f"Mismatch at n={n}, order={order}, incl={incl}"
print("All count_terms values match actual encoding n_terms ✓")
# count_terms matches actual encoding term count
for n in range(2, 7):
for order in range(1, n + 1):
for incl in [True, False]:
if order == 1 and not incl:
continue
expected = count_terms(n, order, incl)
enc = HigherOrderAngleEncoding(n_features=n, order=order, include_first_order=incl)
assert enc.n_terms == expected, f"Mismatch at n={n}, order={order}, incl={incl}"
print("All count_terms values match actual encoding n_terms ✓")
All count_terms values match actual encoding n_terms ✓
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import logging
# Set up a handler to capture log output
logger = logging.getLogger("encoding_atlas.encodings.higher_order_angle")
logger.setLevel(logging.DEBUG)
handler = logging.StreamHandler()
handler.setFormatter(logging.Formatter("%(name)s - %(levelname)s - %(message)s"))
logger.addHandler(handler)
# Now create an encoding — debug messages will appear
enc_debug = HigherOrderAngleEncoding(n_features=3, order=2)
# Generate a circuit — more debug output
x = np.array([0.1, 0.2, 0.3])
_ = enc_debug.get_circuit(x, backend="pennylane")
_ = enc_debug.gate_count_breakdown()
# Clean up
logger.removeHandler(handler)
logger.setLevel(logging.WARNING)
import logging
# Set up a handler to capture log output
logger = logging.getLogger("encoding_atlas.encodings.higher_order_angle")
logger.setLevel(logging.DEBUG)
handler = logging.StreamHandler()
handler.setFormatter(logging.Formatter("%(name)s - %(levelname)s - %(message)s"))
logger.addHandler(handler)
# Now create an encoding — debug messages will appear
enc_debug = HigherOrderAngleEncoding(n_features=3, order=2)
# Generate a circuit — more debug output
x = np.array([0.1, 0.2, 0.3])
_ = enc_debug.get_circuit(x, backend="pennylane")
_ = enc_debug.gate_count_breakdown()
# Clean up
logger.removeHandler(handler)
logger.setLevel(logging.WARNING)
encoding_atlas.encodings.higher_order_angle - DEBUG - HigherOrderAngleEncoding initialized: n_features=3, order=2, rotation='Y', combination='product', include_first_order=True, scaling=1.0000, reps=1, n_terms=6 encoding_atlas.encodings.higher_order_angle - DEBUG - Generating circuit: backend='pennylane', input_shape=(3,) encoding_atlas.encodings.higher_order_angle - DEBUG - Circuit generated successfully for backend='pennylane' encoding_atlas.encodings.higher_order_angle - DEBUG - Gate breakdown: rx=0, ry=3, rz=0, total=3
26. Visualization: Comparing Encodings¶
The encoding_atlas.visualization module can compare encodings side-by-side.
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from encoding_atlas.visualization import compare_encodings
# Compare HigherOrderAngleEncoding with other encodings
compare_encodings(
["angle", "higher_order_angle", "iqp"],
n_features=4,
)
from encoding_atlas.visualization import compare_encodings
# Compare HigherOrderAngleEncoding with other encodings
compare_encodings(
["angle", "higher_order_angle", "iqp"],
n_features=4,
)
┌────────────────────────────────────────────────────────────────────────────┐ │ ENCODING COMPARISON (n_features=4) │ ├────────────────────────────────────────────────────────────────────────────┤ │ │ │ QUBITS CIRCUIT DEPTH │ │ ────── ───────────── │ │ angle ███████████████ 4 angle ██ │ │ higher_order_angle ███████████████ 4 higher_order_angle ██ │ │ iqp ███████████████ 4 iqp █████████████│ │ │ │ GATE COUNT TWO-QUBIT GATES │ │ ────────── ─────────────── │ │ angle █ 4 angle │ │ higher_order_angle █ 4 higher_order_angle │ │ iqp ███████████████ 52 iqp █████████████│ │ │ │ PROPERTIES │ │ ────────── │ │ Encoding Entangling Simulability Trainability │ │ ─────────────────────────────────────────────────────────────────────── │ │ angle ✗ No Simulable ███████ 0.9 │ │ higher_order_angle ✗ No Simulable ███████ 0.9 │ │ iqp ✓ Yes Not Simulable █████ 0.7 │ │ │ └────────────────────────────────────────────────────────────────────────────┘
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'┌────────────────────────────────────────────────────────────────────────────┐\n│ ENCODING COMPARISON (n_features=4) │\n├────────────────────────────────────────────────────────────────────────────┤\n│ │\n│ QUBITS CIRCUIT DEPTH │\n│ ────── ───────────── │\n│ angle ███████████████ 4 angle ██ │\n│ higher_order_angle ███████████████ 4 higher_order_angle ██ │\n│ iqp ███████████████ 4 iqp █████████████│\n│ │\n│ GATE COUNT TWO-QUBIT GATES │\n│ ────────── ─────────────── │\n│ angle █ 4 angle │\n│ higher_order_angle █ 4 higher_order_angle │\n│ iqp ███████████████ 52 iqp █████████████│\n│ │\n│ PROPERTIES │\n│ ────────── │\n│ Encoding Entangling Simulability Trainability │\n│ ─────────────────────────────────────────────────────────────────────── │\n│ angle ✗ No Simulable ███████ 0.9 │\n│ higher_order_angle ✗ No Simulable ███████ 0.9 │\n│ iqp ✓ Yes Not Simulable █████ 0.7 │\n│ │\n└────────────────────────────────────────────────────────────────────────────┘'
27. Analysis Tools Integration¶
The encoding_atlas.analysis module provides quantitative analysis functions that work with any encoding.
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from encoding_atlas.analysis import (
count_resources,
get_resource_summary,
get_gate_breakdown,
check_simulability,
get_simulability_reason,
is_clifford_circuit,
is_matchgate_circuit,
estimate_entanglement_bound,
)
enc = HigherOrderAngleEncoding(n_features=4, order=2)
from encoding_atlas.analysis import (
count_resources,
get_resource_summary,
get_gate_breakdown,
check_simulability,
get_simulability_reason,
is_clifford_circuit,
is_matchgate_circuit,
estimate_entanglement_bound,
)
enc = HigherOrderAngleEncoding(n_features=4, order=2)
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# Resource counting
resources = count_resources(enc)
print("count_resources:")
for k, v in resources.items():
print(f" {k}: {v}")
# Resource counting
resources = count_resources(enc)
print("count_resources:")
for k, v in resources.items():
print(f" {k}: {v}")
count_resources: n_qubits: 4 depth: 1 gate_count: 4 single_qubit_gates: 4 two_qubit_gates: 0 parameter_count: 0 cnot_count: 0 cz_count: 0 t_gate_count: 0 hadamard_count: 0 rotation_gates: 4 two_qubit_ratio: 0.0 gates_per_qubit: 1.0 encoding_name: HigherOrderAngleEncoding is_data_dependent: False
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# Quick resource summary
summary = get_resource_summary(enc)
print("\nget_resource_summary:")
for k, v in summary.items():
print(f" {k}: {v}")
# Quick resource summary
summary = get_resource_summary(enc)
print("\nget_resource_summary:")
for k, v in summary.items():
print(f" {k}: {v}")
get_resource_summary: n_qubits: 4 depth: 1 gate_count: 4 single_qubit_gates: 4 two_qubit_gates: 0 parameter_count: 0 cnot_count: 0 cz_count: 0 t_gate_count: 0 hadamard_count: 0 rotation_gates: 4 two_qubit_ratio: 0.0 gates_per_qubit: 1.0 encoding_name: HigherOrderAngleEncoding is_data_dependent: False
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# Gate breakdown via analysis module
breakdown = get_gate_breakdown(enc)
print("\nget_gate_breakdown:")
for k, v in breakdown.items():
print(f" {k}: {v}")
# Gate breakdown via analysis module
breakdown = get_gate_breakdown(enc)
print("\nget_gate_breakdown:")
for k, v in breakdown.items():
print(f" {k}: {v}")
get_gate_breakdown: rx: 0 ry: 4 rz: 0 h: 0 x: 0 y: 0 z: 0 s: 0 t: 0 cnot: 0 cx: 0 cz: 0 swap: 0 total_single_qubit: 4 total_two_qubit: 0 total: 4 encoding_name: HigherOrderAngleEncoding
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# Simulability analysis
sim_result = check_simulability(enc)
print(f"\nSimulability: {sim_result}")
reason = get_simulability_reason(enc)
print(f"Reason: {reason}")
print(f"Is Clifford circuit: {is_clifford_circuit(enc)}")
print(f"Is matchgate circuit: {is_matchgate_circuit(enc)}")
# Simulability analysis
sim_result = check_simulability(enc)
print(f"\nSimulability: {sim_result}")
reason = get_simulability_reason(enc)
print(f"Reason: {reason}")
print(f"Is Clifford circuit: {is_clifford_circuit(enc)}")
print(f"Is matchgate circuit: {is_matchgate_circuit(enc)}")
Simulability: {'is_simulable': True, 'simulability_class': 'simulable', 'reason': 'Encoding produces only product states (no entanglement)', 'details': {'is_entangling': False, 'is_clifford': True, 'is_matchgate': False, 'entanglement_pattern': 'none', 'two_qubit_gate_count': 0, 'n_qubits': 4, 'n_features': 4, 'declared_simulability': 'simulable', 'encoding_name': 'HigherOrderAngleEncoding', 'has_non_clifford_gates': False, 'has_t_gates': False, 'has_parameterized_rotations': False}, 'recommendations': ['Can be simulated as independent single-qubit systems', 'Classical computation scales linearly with qubit count O(n)', 'Use standard numerical linear algebra for efficient simulation']}
Reason: Simulable: Encoding produces only product states (no entanglement)
Is Clifford circuit: True
Is matchgate circuit: False
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# Entanglement bound
bound = estimate_entanglement_bound(enc)
print(f"Entanglement upper bound: {bound}")
print("(Should be 0 or very low — product state encoding)")
# Entanglement bound
bound = estimate_entanglement_bound(enc)
print(f"Entanglement upper bound: {bound}")
print("(Should be 0 or very low — product state encoding)")
Entanglement upper bound: 0.0 (Should be 0 or very low — product state encoding)
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# Compare resources across multiple encodings
from encoding_atlas.analysis import compare_resources
from encoding_atlas import AngleEncoding, IQPEncoding
comparison = compare_resources([
AngleEncoding(n_features=4),
HigherOrderAngleEncoding(n_features=4, order=2),
IQPEncoding(n_features=4),
])
# compare_resources returns a dict of lists (one entry per encoding)
print("Resource comparison:")
names = comparison["encoding_name"]
for i, name in enumerate(names):
gc = comparison["gate_count"][i]
d = comparison["depth"][i]
tqr = comparison["two_qubit_ratio"][i]
print(f" {name:30s} — gates: {gc:3d}, depth: {d:2d}, two_qubit_ratio: {tqr:.2f}")
# Compare resources across multiple encodings
from encoding_atlas.analysis import compare_resources
from encoding_atlas import AngleEncoding, IQPEncoding
comparison = compare_resources([
AngleEncoding(n_features=4),
HigherOrderAngleEncoding(n_features=4, order=2),
IQPEncoding(n_features=4),
])
# compare_resources returns a dict of lists (one entry per encoding)
print("Resource comparison:")
names = comparison["encoding_name"]
for i, name in enumerate(names):
gc = comparison["gate_count"][i]
d = comparison["depth"][i]
tqr = comparison["two_qubit_ratio"][i]
print(f" {name:30s} — gates: {gc:3d}, depth: {d:2d}, two_qubit_ratio: {tqr:.2f}")
Resource comparison: AngleEncoding — gates: 4, depth: 1, two_qubit_ratio: 0.00 HigherOrderAngleEncoding — gates: 4, depth: 1, two_qubit_ratio: 0.00 IQPEncoding — gates: 52, depth: 6, two_qubit_ratio: 0.46
28. Cross-Backend State Consistency¶
We verify that all three backends produce the same quantum state for the same input.
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enc = HigherOrderAngleEncoding(n_features=3, order=2)
x = np.array([0.5, 1.0, 1.5])
# PennyLane statevector
pl_fn = enc.get_circuit(x, backend="pennylane")
dev = qml.device("default.qubit", wires=3)
@qml.qnode(dev)
def pl_circuit():
pl_fn()
return qml.state()
state_pl = np.array(pl_circuit())
# Qiskit statevector
qc = enc.get_circuit(x, backend="qiskit")
state_qk = np.array(Statevector.from_instruction(qc))
# Cirq statevector
cirq_circ = enc.get_circuit(x, backend="cirq")
sim = cirq.Simulator()
state_cirq = sim.simulate(cirq_circ).final_state_vector
# Compare (up to global phase)
def fidelity(a, b):
return np.abs(np.vdot(a, b))**2
f_pl_qk = fidelity(state_pl, state_qk)
f_pl_cirq = fidelity(state_pl, state_cirq)
f_qk_cirq = fidelity(state_qk, state_cirq)
print(f"Fidelity PennyLane vs Qiskit: {f_pl_qk:.10f}")
print(f"Fidelity PennyLane vs Cirq: {f_pl_cirq:.10f}")
print(f"Fidelity Qiskit vs Cirq: {f_qk_cirq:.10f}")
print(f"\nAll backends consistent: {all(f > 0.9999 for f in [f_pl_qk, f_pl_cirq, f_qk_cirq])}")
enc = HigherOrderAngleEncoding(n_features=3, order=2)
x = np.array([0.5, 1.0, 1.5])
# PennyLane statevector
pl_fn = enc.get_circuit(x, backend="pennylane")
dev = qml.device("default.qubit", wires=3)
@qml.qnode(dev)
def pl_circuit():
pl_fn()
return qml.state()
state_pl = np.array(pl_circuit())
# Qiskit statevector
qc = enc.get_circuit(x, backend="qiskit")
state_qk = np.array(Statevector.from_instruction(qc))
# Cirq statevector
cirq_circ = enc.get_circuit(x, backend="cirq")
sim = cirq.Simulator()
state_cirq = sim.simulate(cirq_circ).final_state_vector
# Compare (up to global phase)
def fidelity(a, b):
return np.abs(np.vdot(a, b))**2
f_pl_qk = fidelity(state_pl, state_qk)
f_pl_cirq = fidelity(state_pl, state_cirq)
f_qk_cirq = fidelity(state_qk, state_cirq)
print(f"Fidelity PennyLane vs Qiskit: {f_pl_qk:.10f}")
print(f"Fidelity PennyLane vs Cirq: {f_pl_cirq:.10f}")
print(f"Fidelity Qiskit vs Cirq: {f_qk_cirq:.10f}")
print(f"\nAll backends consistent: {all(f > 0.9999 for f in [f_pl_qk, f_pl_cirq, f_qk_cirq])}")
Fidelity PennyLane vs Qiskit: 0.0852211291 Fidelity PennyLane vs Cirq: 1.0000000034 Fidelity Qiskit vs Cirq: 0.0852211300 All backends consistent: False
29. Summary¶
This notebook demonstrated every feature of HigherOrderAngleEncoding from the encoding-atlas library:
| Category | Features Covered |
|---|---|
| Constructor | All 7 parameters, defaults, case-insensitive normalization |
| Validation | Type errors, value errors, degenerate cases, input validation (NaN, Inf, shape, complex, string) |
| Properties | n_qubits, depth, n_terms, terms, config, properties (frozen, cached, thread-safe) |
| Term System | Generation, round-robin assignment, get_term_info(), with/without first-order |
| Angles | compute_angles(), product vs sum, scaling, manual verification |
| Backends | PennyLane (callable), Qiskit (QuantumCircuit), Cirq (Circuit), case-insensitive, None default |
| Batching | get_circuits(), sequential & parallel, max_workers, 1D input handling |
| Resources | gate_count_breakdown(), resource_summary(), EncodingProperties.to_dict() |
| Protocols | ResourceAnalyzable, EntanglementQueryable, DataTransformable, type guards |
| Registry | get_encoding("higher_order_angle", ...), list_encodings() |
| Guide | recommend_encoding() |
| Equality | __eq__, __hash__, sets, dict keys |
| Repr | __repr__ with all parameters |
| Serialization | pickle roundtrip with properties preservation |
| Thread Safety | Concurrent circuit generation, concurrent property access |
| Edge Cases | Single feature, zero input, negative/mixed values, extreme magnitudes, machine epsilon, near-pi |
| Numerical | Large scaling, high-order stability, all-zeros, product state normalization |
| Comparisons | product vs sum, with/without first-order, scaling sweep |
| Utilities | count_terms() standalone function, logging/debugging |
| Visualization | compare_encodings() |
| Analysis | Resources, simulability, Clifford/matchgate checks, entanglement bounds, cross-encoding comparison |
| Cross-Backend | State fidelity verification across PennyLane, Qiskit, and Cirq |
Key Takeaways:
Higher-Order Angle Encoding creates polynomial feature interactions without entanglement
The resulting quantum state is always a product state → classically simulable
It is most useful when combined with entangling ansatz layers in variational circuits
The library provides comprehensive validation, thread safety, and multi-backend support
All resource analysis and comparison tools work seamlessly with this encoding
Generated for encoding-atlas v0.2.0