Mathematical Foundations of Quantum Feature Maps

As highlighted in the chapter introduction, translating classical data into the language of quantum mechanics is a necessary first step for quantum machine learning. This translation is formalized through quantum feature maps, functions that embed classical data points into quantum states within a Hilbert space. Understanding the mathematical structure of these maps is fundamental to designing effective QML algorithms.

Defining the Quantum Feature Map

Let's consider a classical dataset where each data point $x$ belongs to some data space $\mathcal{X}$. Often, $\mathcal{X}$ is a subset of $\mathbb{R}^d$, representing $d$-dimensional feature vectors. A quantum feature map, denoted by $\phi$, is a function that maps each classical data point $x \in \mathcal{X}$ to a quantum state $|\phi(x)\rangle$ in a Hilbert space $\mathcal{H}$:

$$\phi: \mathcal{X} \to \mathcal{H}$$

Typically, the Hilbert space $\mathcal{H}$ is the state space of $n$ qubits, so $\mathcal{H} \cong (\mathbb{C}^2)^{\otimes n}$, which has a dimension of $2^n$. The state $|\phi(x)\rangle$ is often called a quantum feature state.

In practice, this mapping is usually implemented via a parameterized quantum circuit, represented by a unitary operator $U_\phi(x)$. This unitary acts on a fixed initial state, commonly the all-zero state $|0\rangle^{\otimes n}$:

$$|\phi(x)\rangle = U_\phi(x) |0\rangle^{\otimes n}$$

Here, the structure of the unitary operator $U_\phi(x)$ depends on the classical input data $x$. The components of $x$ control the parameters of the quantum gates within the circuit $U_\phi(x)$. For example, components of $x$ might determine the rotation angles of single-qubit gates or the strength of multi-qubit interactions.

The Geometry of Quantum Feature Spaces

The feature map $\phi$ effectively embeds the classical data space $\mathcal{X}$ into the high-dimensional quantum state space $\mathcal{H}$. The arrangement and relationships between the quantum feature states $\{|\phi(x)\rangle\}_{x \in \mathcal{X}}$ define the geometry of the quantum feature space. This geometry plays a significant role in the performance of QML algorithms.

A primary way to quantify the relationship between two feature states $|\phi(x)\rangle$ and $|\phi(x')\rangle$ is through their inner product $\langle \phi(x') | \phi(x) \rangle$. The squared absolute value of this inner product,

$$K(x, x') = |\langle \phi(x') | \phi(x) \rangle|^2$$

has a special significance. It corresponds to the probability of measuring the state $|\phi(x')\rangle$ if the system is prepared in state $|\phi(x)\rangle$ (or vice versa). This quantity also defines a positive semi-definite kernel function $K(x, x')$, known as the quantum kernel. This kernel effectively measures the similarity between data points $x$ and $x'$ in the quantum feature space. We will examine quantum kernels in detail in Chapter 3. For now, the important point is that the mathematical structure of the feature map $\phi$ directly dictates the resulting kernel $K$.

Properties of Quantum Feature Maps

Several properties characterize quantum feature maps:

1. High Dimensionality: The dimension of the Hilbert space $\mathcal{H}$ grows exponentially with the number of qubits $n$. This exponential capacity allows feature maps to potentially represent classical data in very high-dimensional spaces, which could be advantageous for finding complex patterns or separating data that is not linearly separable in the original space.
2. Non-linearity: Quantum evolution, especially involving multi-qubit entangling gates controlled by input data $x$, naturally introduces non-linear dependencies. The quantum state $|\phi(x)\rangle$ is generally a highly non-linear function of $x$. This inherent non-linearity is essential for many machine learning tasks.
3. Entanglement: Feature maps can generate entanglement between qubits, depending on the structure of $U_\phi(x)$. Entanglement can create complex correlations in the feature state that might be difficult to capture with classical methods. The ability of a feature map to generate entanglement is often linked to its representational power.

Example: The ZZ Feature Map

A commonly used example is the ZZ feature map. For input data $x \in \mathbb{R}^d$ and an $n$-qubit system (typically $n=d$), one popular form is:

$$U_\phi(x) = U_{\text{ent}} H^{\otimes n} U_P(x) H^{\otimes n}$$

Where:

• $H^{\otimes n}$ applies a Hadamard gate to each qubit, creating an initial superposition.
• $U_P(x) = \exp\left(i \sum_{j=1}^n x_j Z_j\right)$ encodes the components $x_j$ of the input vector $x$ into rotations around the Z-axis for each qubit $j$. $Z_j$ is the Pauli-Z operator acting on qubit $j$.
• $U_{\text{ent}} = \exp\left(i \sum_{j, k \text{ connected}} \phi(x_j, x_k) Z_j Z_k\right)$ introduces entanglement based on interactions between connected pairs of qubits. The function $\phi(x_j, x_k)$ depends on the input features $x_j$ and $x_k$. A common choice is $\phi(x_j, x_k) = (\pi - x_j)(\pi - x_k)$.

The circuit applies layers of single-qubit rotations dependent on $x$ interleaved with entangling gates. The exact form of $U_P(x)$ and $U_{\text{ent}}$ can vary. For instance, the circuit might be repeated multiple times ($L$ layers) to increase complexity:

$$U_\phi(x) = \left( U_{\text{ent}} U_P(x) H^{\otimes n} \right)^L |0\rangle^{\otimes n}$$

Here's a circuit diagram for a 2-qubit ZZ feature map with one layer ($L=1$) and interaction $\phi(x_1, x_2) = (\pi - x_1)(\pi - x_2)$:

A 2-qubit ZZ Feature Map circuit. Input data $x=(x_1, x_2)$ controls the rotation angles of $RZ$ gates and the strength of the $ZZ$ interaction term (implemented using CNOTs and an $RZ$ gate). $H$ represents the Hadamard gate.

This example shows how classical data $x$ directly parameterizes the unitary evolution $U_\phi(x)$, creating a specific quantum state $|\phi(x)\rangle$ for each input $x$.

From Pure States to Density Matrices

While we often discuss feature maps generating pure states $|\phi(x)\rangle$, the framework can be extended to map classical data to mixed states, represented by density matrices $\rho(x)$. This might occur naturally when considering noise effects during the state preparation $U_\phi(x)$ or if the encoding strategy itself is designed to produce mixed states. Recalling the formalism from Chapter 1, the feature map becomes $\phi: \mathcal{X} \to \mathcal{D}(\mathcal{H})$, where $\mathcal{D}(\mathcal{H})$ is the space of density operators on $\mathcal{H}$. The kernel definition can also be adapted for density matrices using metrics like the Hilbert-Schmidt inner product or fidelity.

In summary, the mathematical foundation of quantum feature maps lies in the definition of a mapping $\phi$ from a classical data space $\mathcal{X}$ to a quantum Hilbert space $\mathcal{H}$, typically implemented by a data-dependent unitary $U_\phi(x)$. This mapping embeds classical data into a high-dimensional quantum state space, defining a geometry characterized by inner products (kernels). The properties of this map, such as its non-linearity and entangling capability, are determined by the structure of $U_\phi(x)$ and are central to the potential advantages of QML models. The subsequent sections will build upon this foundation to explore specific types of feature maps and methods for analyzing their capabilities.