Quantum Data Representation in Quantum Machine Learning

Introduction

Quantum data representation is a foundational concept in quantum machine learning. Unlike classical data, which is encoded as bits, quantum data is represented using qubits. Qubits enable more expressive and high-dimensional data encoding through principles such as superposition, entanglement, and interference. This page introduces the essential methods for encoding classical data into quantum states—an indispensable step for developing any quantum AI model.

What Makes Quantum Data Unique?

Superposition: A qubit can exist in multiple states simultaneously, allowing quantum systems to represent and process more information in parallel than classical bits.
Entanglement: Qubits can be interdependent in such a way that the state of one affects another, enabling the encoding of complex relationships.
Amplitude Encoding: Data is embedded in the amplitudes of a quantum state, providing a compact way to represent large datasets.
Interference: Quantum computations exploit interference to control probabilities and amplify correct outcomes while canceling out incorrect ones.

Core Quantum Encoding Techniques

1. Basis Encoding

Encodes classical binary data directly into the computational basis states of qubits.
Efficient for binary or categorical data.
Does not leverage superposition but is simple to implement.

Example: The binary string “10” is encoded as $|10\rangle$ using two qubits.

2. Amplitude Encoding

Encodes numerical vectors into the amplitude of quantum states.
Very efficient: $n$ qubits can represent $2^n$ data points.
Requires the input vector to be normalized.
Complex to prepare on current quantum devices.

Example: A normalized vector $[a, b, c, d]$ is encoded as $a|00\rangle + b|01\rangle + c|10\rangle + d|11\rangle$.

3. Angle Encoding (Rotation Encoding)

Maps each classical feature to a rotation angle using gates like RX, RY, or RZ.
Ideal for use in variational quantum circuits (VQCs).
Hardware-friendly and resilient to noise.

Example: A feature $x$ can be encoded using $RY(x)|0\rangle$.

4. QSample Encoding

Encodes probability distributions into quantum states.
Suited for models requiring probabilistic behavior (e.g., Quantum Boltzmann Machines).
Involves sampling classical data to generate quantum state distributions.

Advanced Encoding Strategies

Time-Based Encoding

Data is encoded as the evolution time of a quantum system under a Hamiltonian.
Common in analog quantum computing and certain quantum simulations.

Entangled Feature Encoding

Involves creating entanglement among qubits during encoding to capture feature correlations.
Enhances expressiveness of quantum circuits but increases design complexity.

How to Choose an Encoding Method

Selecting an encoding strategy depends on several factors:

Dimensionality: Use amplitude encoding for very large datasets.
Circuit Depth and Resources: Prefer simple encodings like angle encoding for NISQ devices.
Hardware Constraints: Some encodings require deep or complex circuits unsuitable for current devices.
Model Compatibility: Choose encodings that align with your quantum model type—VQCs work well with angle encoding.

PennyLane Example: Angle Encoding

Below is a simple PennyLane implementation of angle encoding using the RY gate:

import pennylane as qml
from pennylane import numpy as np

def angle_encoding(data):
    for i in range(len(data)):
        qml.RY(data[i], wires=i)

This function rotates each qubit by an angle corresponding to a data feature.

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