Project: A Quantum Classifier

Quantum machine learning models are parameterized circuits trained like neural networks: a feature is encoded into a quantum state, a trainable unitary transforms it, and a measurement reads off the prediction. This capstone builds the smallest non-trivial example — a single-qubit data-reuploading classifier — and connects training, inference, and benchmarking.

Encoding and the model

A scalar feature $x$ is loaded by a rotation, $|\phi(x)\rangle = R_y(x)|0\rangle$ . A trained weight $w$ is applied as a second rotation, and the label is read from a $Z$ -measurement. Because both rotations share the $Y$ axis,

|\psi(x, w)\rangle = R_y(w)\,R_y(x)\,|0\rangle = R_y(x + w)\,|0\rangle = \cos\!\Big(\tfrac{x+w}{2}\Big)|0\rangle + \sin\!\Big(\tfrac{x+w}{2}\Big)|1\rangle.

The decision rule is a threshold on the measured probability:

\hat{y}(x) = \begin{cases} 0 & P(|0\rangle) > P(|1\rangle), \\ 1 & \text{otherwise.}\end{cases}

Training

Training minimizes a loss — e.g. the squared error between $P(|1\rangle)$ and the true label — averaged over the dataset, by adjusting $w$ with a classical optimizer using parameter-shift gradients. For a threshold dataset where points with $x > 0$ belong to class $1$ , the optimizer converges to $w = \pi/2$ , which rotates positive- $x$ points past the equator of the Bloch sphere toward $|1\rangle$ .

Inference and benchmarking

At the converged weight, classify the test point $x = \pi/2$ :

R_y(\pi/2 + \pi/2)\,|0\rangle = R_y(\pi)\,|0\rangle = |1\rangle,

a confident class- $1$ prediction with $P(|1\rangle) = 1$ . Benchmarking the model means sweeping the test set, recording $\hat{y}$ for each point, and reporting accuracy — the same protocol used for any classifier, quantum or classical.

Try it

Build the classifier for the test point $x = \pi/2$ at the trained weight $w = \pi/2$ and measure. The grader checks the output distribution; all probability should land on outcome $1$ .

Run your code to see the quantum state.