Encoding High-Dimensional Classical Data

Classical machine learning frequently deals with datasets where each data point is represented by a vector with a large number of features, often thousands or even millions. Think of image recognition, natural language processing, or genomic analysis. Bringing such high-dimensional data into the quantum domain presents a significant challenge, often referred to as the "input problem" in QML. Simply mapping each feature to a qubit (like in basic basis encoding) is infeasible with current and near-term quantum hardware, which typically offers only tens to hundreds of qubits. We need smarter strategies to represent this rich classical information compactly and effectively within the constraints of available quantum resources.

This section examines techniques and considerations for encoding high-dimensional classical data vectors $x \in \mathbb{R}^N$, where $N$ is large, into quantum states $|\phi(x)\rangle$ using a manageable number of qubits, $n$, where ideally $n \ll N$.

The Challenge of Dimensionality

The most direct, information-preserving way to encode a vector $x = (x_1, x_2, ..., x_N)$ might seem to be amplitude encoding. Here, the $N=2^n$ features are normalized and encoded as the amplitudes of a quantum state on $n$ qubits:

$$|\phi(x)\rangle = \frac{1}{\|x\|} \sum_{i=1}^{N} x_i |i\rangle$$

where $|i\rangle$ represents the computational basis state corresponding to the binary representation of $i$. This achieves an exponential compression, encoding $2^n$ features using only $n$ qubits. However, preparing an arbitrary quantum state like $|\phi(x)\rangle$ is generally difficult. Known algorithms often require circuits with a depth that scales polynomially or even exponentially with $n$, potentially negating the quantum advantage sought. Furthermore, extracting information about specific features (amplitudes) from the resulting state is non-trivial.

Basis encoding, where each feature might control the state of a specific qubit or group of qubits, scales linearly with the number of features ($N$ features need $\mathcal{O}(N)$ qubits), making it impractical for high-dimensional data on current hardware.

Angle encoding, mapping features $x_i$ to rotation angles $\theta_i(x_i)$ applied to individual qubits (e.g., $R_X(\theta_i(x_i))$, $R_Y(\theta_i(x_i))$, $R_Z(\theta_i(x_i))$), often requires $n \approx N$ qubits if each feature gets its own rotation gate. While variations exist using fewer qubits with more complex circuits (like dense angle encoding or data re-uploading), the fundamental challenge remains: how do we condense high-dimensional information effectively?

Strategies for High-Dimensional Encoding

Facing these limitations, several strategies are employed, often in combination:

1. Classical Dimensionality Reduction

Before even considering quantum encoding, applying classical dimensionality reduction techniques is often the most practical first step. If the intrinsic dimensionality of the data is much lower than the ambient dimension $N$, techniques like:

• Principal Component Analysis (PCA): Finds orthogonal directions of maximum variance in the data. Projecting the data onto the top $k$ principal components can significantly reduce dimensionality while retaining most of the variance.
• Autoencoders: Neural networks trained to compress data into a low-dimensional latent representation (encoding) and then reconstruct it (decoding). The compressed latent vector can then be encoded onto qubits.
• Feature Selection: Algorithms that identify and select the most informative subset of the original features based on statistical properties or model performance.

The goal is to obtain a lower-dimensional representation $x' \in \mathbb{R}^{N'}$ with $N' \ll N$, which is then easier to encode onto $n \approx N'$ or even fewer qubits using quantum methods. This preprocessing step uses powerful classical ML tools and focuses the quantum resources on the most salient information.

2. Qubit-Efficient Quantum Encoding Schemes

Even after classical reduction, $N'$ might still be too large for naive encoding. Qubit-efficient quantum encoding strategies aim to use $n < N'$ qubits:

• Amplitude Encoding (Revisited): Despite its state preparation challenges, if $N' = 2^n$, it remains the most qubit-efficient method. Research is ongoing into efficient preparation methods for specific classes of states relevant to ML tasks. It's particularly relevant if the subsequent quantum algorithm can process the information holistically in the amplitudes without needing to access them individually.
• Angle Encoding Variations:
  • Dense Angle Encoding: Multiple features ($k$) can be encoded onto a single qubit using sequences of rotations around different axes, e.g., $R_Z(\theta_1) R_Y(\theta_2) R_Z(\theta_3)...$ where $\theta_j$ depends on some subset of features. If $k$ features are encoded per qubit, $N'$ features require roughly $n \approx N'/k$ qubits.
  • Data Re-uploading: As discussed previously, circuits employing data re-uploading can process high-dimensional data using a fixed, potentially small, number of qubits ($n$). The classical data $x$ influences multiple parameterized gates throughout the circuit. This implicitly creates a complex feature map without needing $N'$ qubits. The expressivity depends on the circuit depth and structure rather than solely on the number of qubits.

Typical workflow involving classical dimensionality reduction before quantum encoding for high-dimensional data.

3. Random Projections

Inspired by classical techniques like the Johnson-Lindenstrauss lemma, random projections can map high-dimensional data to a lower-dimensional space while approximately preserving distances. Quantum equivalents involve applying random (or pseudo-random) quantum circuits parameterized by the data. While potentially efficient in terms of circuit depth, analyzing the properties of such random feature maps is an active area of research.

Trade-offs

Choosing an encoding strategy for high-dimensional data involves navigating several trade-offs:

• Qubit Count: How many qubits ($n$) are required relative to the data dimension ($N$ or $N'$)? Amplitude encoding is optimal here, while basis encoding is the worst. Angle encoding variants fall in between.
• Circuit Depth/Complexity: How many gates are needed to implement the encoding? Amplitude encoding can be deep, while simple angle encoding is shallow but qubit-hungry. Data re-uploading offers a trade-off between qubit count and depth. NISQ hardware favors shallow circuits.
• Information Loss: Classical dimensionality reduction inherently involves some information loss. Quantum encoding methods might also implicitly discard information or make it inaccessible depending on the structure.
• Expressivity and Entanglement: Does the encoding create a rich feature space? Simple product states from basic angle encoding might lack the expressivity needed for complex tasks. More complex encodings or those using entanglement can create richer feature maps but are harder to implement and analyze.
• Algorithm Compatibility: The choice of encoding impacts subsequent algorithms. Kernel methods rely on efficiently calculating inner products $\langle\phi(x)|\phi(x')\rangle$. VQAs rely on the structure of $|\phi(x)\rangle$ and how parameters affect measurements. The encoding must be compatible with how the quantum algorithm processes information.

Effectively encoding high-dimensional data remains a critical bottleneck and an active research area in QML. Often, a hybrid approach combining classical preprocessing with a carefully chosen, qubit-efficient quantum encoding scheme provides the most viable path forward, especially when targeting near-term quantum devices. The optimal strategy is typically data-dependent and task-specific, requiring experimentation and careful analysis of these trade-offs.