Variational Classifiers

A variational quantum classifier (VQC) is a parameterized circuit trained to assign labels. It combines the feature map of the kernel lesson with a trainable ansatz, then reads out a prediction from a measured expectation value. Mitarai et al. (2018) and Havlíček et al. (2019) introduced this "quantum circuit learning" template.

Anatomy of the model

The circuit has three stages applied to $|0\rangle^{\otimes n}$ :

|\psi(\boldsymbol{x},\boldsymbol{\theta})\rangle = W(\boldsymbol{\theta})\,U_\Phi(\boldsymbol{x})\,|0\rangle^{\otimes n}.

Encoding $U_\Phi(\boldsymbol{x})$ — the fixed feature map that loads the data (e.g. angle encoding $R_Y(2x_i)$ ).
Ansatz $W(\boldsymbol{\theta})$ — trainable rotations (and entangling gates for more than one qubit) whose parameters are learned.
Readout — measure an observable, typically $\langle Z\rangle$ on one qubit, and map its sign to a class.

The prediction is

f(\boldsymbol{x}) = \langle \psi(\boldsymbol{x},\boldsymbol{\theta})|\,Z\,|\psi(\boldsymbol{x},\boldsymbol{\theta})\rangle \in [-1, 1],

and the class is $\operatorname{sign} f(\boldsymbol{x})$ .

A one-qubit example

The smallest non-trivial VQC uses a single qubit. With a scalar feature $x$ encoded by $R_Y(2x)$ and a two-angle trainable block $R_Y(\theta_0)\,R_Z(\theta_1)$ , the prepared state is

|\psi\rangle = R_Y(\theta_0)\,R_Z(\theta_1)\,R_Y(2x)\,|0\rangle.

Even this tiny model can separate simple 1-D data: training rotates the Bloch vector so that points of one class land in the upper hemisphere ( $\langle Z\rangle > 0$ ) and the other class in the lower hemisphere.

Training the parameters

The classical loop minimizes a loss — for binary labels $y\in\{-1,+1\}$ , a common choice is the squared error $\bigl(f(\boldsymbol{x}) - y\bigr)^2$ averaged over the dataset. Gradients of $f$ with respect to each $\theta_k$ come from the parameter-shift rule (Module 7): for a rotation generated by a Pauli,

\frac{\partial f}{\partial \theta_k} = \tfrac12\Bigl[f(\theta_k + \tfrac{\pi}{2}) - f(\theta_k - \tfrac{\pi}{2})\Bigr].

These exact analytic gradients feed an ordinary optimizer (gradient descent, Adam) that updates $\boldsymbol{\theta}$ .

Try it

Prepare the state of a one-qubit variational classifier for the input $x = 0.5$ with the trained parameters $\theta_0 = 0.8$ (an $R_Y$ ) and $\theta_1 = 0.3$ (an $R_Z$ ). Apply the gates in the order encoding, then $R_Z(\theta_1)$ , then $R_Y(\theta_0)$ — i.e. build

R_Y(0.8)\,R_Z(0.3)\,R_Y(2\cdot 0.5)\,|0\rangle.

The grader checks the resulting state vector.

Run your code to see the quantum state.