Parameterized Quantum Circuits

A parameterized quantum circuit (PQC) — also called a variational ansatz — is a quantum circuit whose gate angles are continuous free parameters $\boldsymbol{\theta} \in \mathbb{R}^p$ . Rather than being fixed at design time, those angles are adjusted by a classical optimizer to minimize a cost function evaluated by running the circuit on a quantum device (or simulator).

This hybrid classical-quantum loop is the backbone of the Variational Quantum Eigensolver (VQE), the Quantum Approximate Optimization Algorithm (QAOA), and many Quantum Neural Networks (QNNs).

Anatomy of an ansatz

A typical hardware-efficient ansatz repeats a pattern of single-qubit rotations followed by entangling gates for $L$ layers:

U(\boldsymbol{\theta}) = \prod_{\ell=1}^{L} \Bigl( \bigotimes_{j} R_y(\theta_{\ell,j}) \Bigr) \cdot E_\ell,

where $R_y(\theta)$ is the $Y$ -rotation and $E_\ell$ is a fixed entangling layer (often a chain of CNOTs). For $n$ qubits and $L$ layers the circuit has $nL$ free parameters.

The Ry gate

The single-qubit $Y$ -rotation by angle $\theta$ is

R_y(\theta) = e^{-i\theta Y/2} = \begin{pmatrix} \cos\tfrac{\theta}{2} & -\sin\tfrac{\theta}{2} \\ \sin\tfrac{\theta}{2} & \cos\tfrac{\theta}{2} \end{pmatrix}.

It maps $|0\rangle \mapsto \cos(\theta/2)|0\rangle + \sin(\theta/2)|1\rangle$ , so at $\theta = \pi$ it acts as an $X$ gate; at $\theta = \pi/2$ it prepares an equal superposition with real, positive amplitudes.

Why Ry?

Unlike $R_z$ , an $R_y$ rotation moves amplitude between $|0\rangle$ and $|1\rangle$ with real coefficients, which keeps the state on the $XZ$ great-circle of the Bloch sphere. This makes gradients easier to estimate and the landscape more amenable to classical optimizers.

Building a trainable model

For a 2-qubit, single-layer ansatz the circuit looks like:

|\psi(\theta_0, \theta_1)\rangle = \text{CNOT}_{01} \cdot \bigl(R_y(\theta_0) \otimes R_y(\theta_1)\bigr)|00\rangle.

Choosing $\theta_0 = \pi/3$ and $\theta_1 = \pi/4$ gives concrete amplitudes we can compute:

After the $R_y$ rotations: $\cos(\pi/6)|0\rangle + \sin(\pi/6)|1\rangle$ on qubit 0, and $\cos(\pi/8)|0\rangle + \sin(\pi/8)|1\rangle$ on qubit 1.
The CNOT then flips qubit 1 conditional on qubit 0, creating entanglement between the two qubits.

The parameter-shift rule

Because $U(\boldsymbol{\theta})$ depends on rotation angles, the gradient of any expectation value $\langle O \rangle(\theta)$ can be computed analytically on hardware:

\frac{\partial \langle O \rangle}{\partial \theta_j} = \tfrac{1}{2}\bigl[\langle O \rangle(\theta_j + \tfrac{\pi}{2}) - \langle O \rangle(\theta_j - \tfrac{\pi}{2})\bigr].

This parameter-shift rule (Mitarai et al., 2018; Schuld et al., 2019) requires exactly two circuit evaluations per parameter and works on real quantum hardware — no finite-difference approximation is needed.

Try it

Build the 2-qubit single-layer ansatz: apply $R_y(\pi/3)$ to qubit 0, $R_y(\pi/4)$ to qubit 1, then entangle them with a CNOT. The grader checks the exact statevector produced by your circuit.

Run your code to see the quantum state.