QML Landscape

Quantum machine learning (QML) is not a single algorithm but a family of strategies for using a parameterized quantum circuit as a trainable model. This module surveys the advanced end of that family: kernel methods, variational classifiers, generative models, and the trainability and advantage questions that decide whether any of it pays off. This first lesson lays out the map so the later lessons fit together.

Two complementary viewpoints

Most near-term QML reduces to one of two pictures, and it helps to hold both.

The variational (model) view. A quantum circuit $U(\boldsymbol{x}, \boldsymbol{\theta})$ takes a data point $\boldsymbol{x}$ and trainable parameters $\boldsymbol{\theta}$ , prepares a state $|\psi(\boldsymbol{x},\boldsymbol{\theta})\rangle$ , and a measured observable produces a prediction $f(\boldsymbol{x}) = \langle \psi | M | \psi \rangle$ . Training adjusts $\boldsymbol{\theta}$ to minimize a loss — exactly the variational quantum algorithm pattern from Module 7, now applied to a learning task rather than a ground-state search.

The kernel (feature-map) view. Encoding a data point with a fixed circuit $\Phi: \boldsymbol{x} \mapsto |\phi(\boldsymbol{x})\rangle$ embeds it in the exponentially large Hilbert space of the qubits. Inner products in that space,

k(\boldsymbol{x}, \boldsymbol{x}') = \bigl|\langle \phi(\boldsymbol{x}) | \phi(\boldsymbol{x}') \rangle\bigr|^2,

define a kernel that a classical support-vector machine can train on. Schuld and Killoran showed the two views are deeply linked: a variational model with a fixed feature map is a linear model in the quantum feature space.

The pieces of a QML model

Every model in this module is built from the same four ingredients:

Encoding (feature map): how classical data enters the circuit — angle encoding, amplitude encoding, or repeated data re-uploading.
Ansatz: the trainable layers, chosen for expressibility while avoiding barren plateaus.
Measurement / readout: the observable whose expectation becomes the output.
Loss and optimizer: the classical part of the hybrid loop that updates $\boldsymbol{\theta}$ .

What the rest of the module covers

| Lesson | Question it answers | | --- | --- | | Quantum kernels | When does a quantum feature map give a useful similarity measure? | | Variational classifiers | How do we train a circuit to label data? | | Generative models / QGANs | Can a circuit sample from a learned distribution? | | Data re-uploading | How does repeating the encoding boost expressivity? | | Trainability & barren plateaus | Why do gradients vanish, and how do we keep them alive? | | Quantum advantage in learning | Where is there a provable or plausible separation? | | Benchmarking & hybrid architectures | How do we measure success honestly and combine with classical nets? |

A note of honesty up front

The QML literature has a healthy dose of hype. The recurring theme of this module is that encoding choices, trainability, and fair benchmarking matter more than circuit cleverness. A model with a beautiful ansatz but vanishing gradients learns nothing; a quantum kernel that is efficiently simulable classically offers no advantage. Keeping these caveats in view is what separates an advanced practitioner from a demo. The variational-algorithm framework of Cerezo et al. and the feature-space framing of Schuld and Killoran are the two lenses we will keep returning to.