Now that we've explored different ways to construct quantum feature maps, $\phi: x \mapsto |\phi(x)\rangle$ , a natural question arises: How effective are these maps? Not all encoding strategies are created equal. Their ability to transform classical data into quantum states with useful properties directly impacts the performance of downstream quantum machine learning algorithms. Two fundamental properties govern this effectiveness: expressibility and entangling capability. Understanding and quantifying these aspects allows us to design better feature maps tailored to specific problems and hardware.

Understanding Expressibility

Expressibility refers to the extent to which a quantum feature map can generate diverse quantum states across the Hilbert space as the input data $x$ varies. Think of it as the "coverage" of the feature space. A highly expressive feature map can produce states populating a large volume or a complex manifold within the state space, while a map with low expressibility might confine all encoded data points to a small or simple region.

Why does this matter? The expressibility of the feature map fundamentally limits the complexity of the functions that a subsequent quantum model (like a quantum kernel machine or a variational classifier) can represent. If the feature map doesn't sufficiently separate or structure the data points in the Hilbert space, even the most powerful quantum algorithm built upon it might fail to find a good decision boundary or model the underlying data distribution effectively.

Quantifying Expressibility

Quantifying expressibility is an active area of research, but several approaches exist. One common idea is to compare the distribution of states $\{|\phi(x)\rangle\}$ generated by the feature map (for a representative set of inputs $x$ ) to a uniform distribution over the Hilbert space, often approximated by the Haar measure.

Numerically, this can involve:

Sampling: Generate many states $|\phi(x_i)\rangle$ by feeding sample data points $x_i$ into the feature map circuit $\mathcal{U}_{\phi(x)}$ .
State Tomography (Limited): For small systems, one could try to reconstruct the density matrices $\rho(x_i) = |\phi(x_i)\rangle\langle\phi(x_i)|$ .
Comparing Distributions: Calculate moments (like the average purity or higher-order moments) of the sampled state distribution and compare them to the moments expected from the Haar random distribution. Significant deviations indicate limited expressibility.

Another approach involves analyzing the algebraic structure generated by the gates in the feature map circuit, connecting expressibility to concepts from Lie algebra and control theory. High expressibility is often associated with the feature map circuit generating a Lie algebra that spans the entire space of operators (like $\mathfrak{su}(2^n)$ for $n$ qubits).

However, maximum expressibility isn't always desirable. Highly expressive circuits can be difficult to train, potentially leading to issues like barren plateaus, which we will discuss in Chapter 4. There's often a trade-off between expressibility and trainability.

Gauging Entangling Capability

Entanglement, the uniquely quantum correlation between subsystems, is often cited as a potential source of quantum advantage in QML. The entangling capability of a feature map measures its tendency to generate entangled states $|\phi(x)\rangle$ . A feature map consisting only of single-qubit rotations, for instance, would have zero entangling capability, as it cannot create entanglement between qubits.

The ability to generate entanglement is significant because it allows the quantum state to encode correlations between input features in ways that are classically intractable for high dimensions. This might be particularly beneficial for datasets where complex feature interactions are important.

Measuring Entanglement

Several measures can quantify the entanglement generated by a feature map. A practical approach often involves computing an entanglement measure for the output state $|\phi(x)\rangle$ and averaging it over a dataset $\{x_i\}$ . Common entanglement measures for pure states include:

Entanglement Entropy: For a bipartite split of the qubits (say, set A and set B), calculate the von Neumann entropy of the reduced density matrix of one subsystem: $S(\rho_A) = -\text{Tr}(\rho_A \log_2 \rho_A)$ , where $\rho_A = \text{Tr}_B(|\phi(x)\rangle\langle\phi(x)|)$ . Averaging $S(\rho_A)$ over different inputs $x$ gives a sense of the typical entanglement across that specific bipartition.
Meyer-Wallach Measure: This measure, $Q(|\psi\rangle)$ , quantifies the average entanglement between each qubit and the rest. It's defined as $Q(|\psi\rangle) = \frac{1}{n} \sum_{k=1}^n L_k$ , where $L_k$ is the linear entropy $L_k = 2(1 - \text{Tr}(\rho_k^2))$ and $\rho_k$ is the reduced density matrix of the $k$ -th qubit. $Q$ ranges from 0 (product state) to 1 (maximally entangled state average).

The entangling capability is directly related to the presence and placement of multi-qubit gates (like CNOT or CZ) within the feature map circuit. Maps employing more entangling gates, or arranging them in specific patterns, generally exhibit higher entangling capability.

A simple 2-qubit feature map circuit. Single-qubit rotations $R_x, R_y, R_z$ encode data $x = (x_0, x_1)$ and potentially trainable parameters $p_0, p_1$ . The CNOT gate (CX) is responsible for generating entanglement. The entangling capability depends on the input $x$ and parameters $p$ .

The Interplay: Expressibility vs. Entanglement

While distinct, expressibility and entangling capability are often intertwined. Feature maps capable of generating significant entanglement typically explore larger regions of the Hilbert space, contributing to higher expressibility. A circuit lacking entangling gates can only generate product states, severely limiting its expressibility to a small submanifold of the full Hilbert space.

However, high entangling power doesn't automatically guarantee high expressibility relevant for a specific task. It's possible to generate highly entangled states that are clustered in a way that doesn't effectively separate different classes of data. Conversely, some level of expressibility can be achieved even with limited entanglement, particularly in how single-qubit states are distributed.

The ideal balance depends on the problem. For tasks believed to require capturing complex, non-local correlations, prioritizing entangling capability might be beneficial. For others, ensuring sufficient coverage (expressibility) of the relevant part of the feature space might be more important, potentially with simpler, less entangled structures to aid trainability.

Practical Considerations

Analyzing these properties helps guide the design of feature map circuits:

Circuit Depth and Structure: Deeper circuits with alternating layers of data encoding rotations and entangling blocks tend to increase both expressibility and entangling capability, but also increase susceptibility to noise and training difficulties.
Gate Choice: The type of entangling gates (e.g., CNOT vs. controlled-phase) and their placement influences the nature of correlations generated.
Data Dependence: Remember that the actual state $|\phi(x)\rangle$ and its properties depend on the input data $x$ . Analysis often involves averaging over expected data distributions.

Tools within quantum computing libraries like PennyLane provide functionalities to compute quantities related to expressibility (e.g., using the qml.expressibility function, which often relies on comparing state moments to Haar random) and entanglement (e.g., qml.vn_entropy, qml.purity). These allow for empirical evaluation of different feature map designs.

Understanding the expressibility and entangling capability of your chosen feature map is not just a theoretical exercise. It provides insights into the potential power and limitations of your QML model, guiding architecture choices and helping to interpret results, especially when comparing quantum approaches to classical benchmarks or when diagnosing training problems. As we move towards quantum kernel methods in the next chapter, we'll see how the geometry of the feature space, shaped by these properties, directly defines the quantum kernel.