In order to answer your question, we need to know:

1. What is a quantum feature map?

2. Where and how a quantum feature map is used in QML?

3. What is the difference between different entanglement layer structures `linear`

vs `full`

?

4. How a quantum feature map is realized in a real quantum hardware?

5. Run time comparison for `linear`

and `full`

entanglement layer?

**1. What is a quantum feature map?**

For the background knowledge, quantum feature maps are widely used in the process of quantum embedding. This process **encodes classical data $\vec{x}$ into quantum data $|\psi(\vec{x})\rangle = U(\vec{x})|0\rangle^{\otimes n}$** via a parametrized quantum circuit (PQC) $U(\vec{x})$ (in the context of QML). For a brief review, check Pennylane's post. Notably, the authors discuss the relation between encoding data multiple times and the universality of QML models in ref [1] by considering the accessible frequencies $\Omega$ in a partial Fourier series $f(\vec{x}) = \sum_{\omega \in \Omega} c_\omega e^{i\omega x}$.

For the frequently used quantum feature maps, please check:

- Embedding templates from Pennylane

- Qiskit Circuit Library -> Data encoding circuits

**Note:** Similar to the choice of variational ansatz in VQE, *there is no general recipe for choosing the best quantum feature map*!! Unfortunately, most QML algorithms are heuristics which means you have to tune the model parameters and see how it actually behaves. One can also design a meta-learning algorithm to automatically construct better quantum feature maps [2].

**2. Where and how a quantum feature map is used in QML?**

It is usually used in quantum supervised learning like a Variational Quantum Classifier and Quantum Kernel Methods. From my experience, a `full`

entanglement layer **does not always give better perdition accuracy** in quantum classifiers. But the idea behind is **a complicated entangling layer is hard to simulate classically which could bring potential quantum advantage**.

**3. What is the difference between different entanglement layer structures **`linear`

vs `full`

?

According to Qiskit's API on the TwoLocal circuit:

- `full`

entanglement is each qubit is **entangled with all the others**.

- `linear`

entanglement is qubit 𝑖 entangled with qubit 𝑖+1, for all 𝑖∈{0,1,...,𝑛−2}, where 𝑛 is the total number of qubits. Also known as the hardware-efficient ansatz [3].

- `circular`

entanglement is linear entanglement but with an additional entanglement of the first and last qubit.

For example, considering the following 4-qubit quantum circuit with `full`

/`linear`

/`circular`

entanglement:

**4. How a quantum feature map is realized in a real quantum hardware?**

Similar to other quantum circuits, the feature map eventually will be transpiled into physical signals (microwave pulses for superconducting qubits). The issue happens when considering a `full`

entanglement layer. Since some physical qubits are not connected with each other (limited by the topology/coupling map of a specific quantum device), **we cannot directly apply two qubit gates on those qubits and it will take a longer time to go through the transpiler to resolve this by inserting SWAP gates**. This is usually known as the **Qubit Mapping Problem** for NISQ devices [4].

**Note:** For this device, $q_0$ and $q_9$ are not connected so we cannot directly apply two qubit gates on them.

**5. Run time comparison for **`linear`

and `full`

entanglement layer?

Since there are usually more two qubit gates in the `full`

entanglement layer than the `linear`

structure, the former naturally takes a longer run time. Besides, too much entanglement in the QML model will also induce the notorious Barren Plateau issue [5].

**Reference:**

[1] Schuld, Maria, Ryan Sweke, and Johannes Jakob Meyer. "Effect of data encoding on the expressive power of variational quantum-machine-learning models." *Physical Review A* 103.3 (2021): 032430.

[2] Altares-López, Sergio, Angela Ribeiro, and Juan José García-Ripoll. "Automatic design of quantum feature maps." *arXiv preprint arXiv:2105.12626* (2021).

[3] Kandala, Abhinav, et al. "Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets." *Nature* 549.7671 (2017): 242-246.

[4] Li, G., Y. Ding, and Y. Xie. "Tackling the Qubit Mapping Problem for NISQ-Era Quantum Devices. arXiv e-prints." *arXiv preprint arXiv:1809.02573* (2018).

[5] Marrero, Carlos Ortiz, Mária Kieferová, and Nathan Wiebe. "Entanglement induced barren plateaus." *arXiv preprint arXiv:2010.15968* (2020).