Cost Functions and Gradients

Training a variational quantum circuit means picking parameters (gate angles $\boldsymbol{\theta}$ ) so that a cost function $C(\boldsymbol{\theta})$ is minimised. The classical optimizer that adjusts those angles relies on gradients — derivatives $\partial C / \partial \theta_j$ that indicate which direction to move each angle.

Expectation values as cost functions

The natural cost function in quantum computing is the expectation value of an observable $O$ in the parametric state $|\psi(\boldsymbol{\theta})\rangle$ :

C(\boldsymbol{\theta}) = \langle O \rangle(\boldsymbol{\theta}) \;=\; \langle \psi(\boldsymbol{\theta}) | O | \psi(\boldsymbol{\theta}) \rangle.

Because $C$ is real-valued and bounded, it is a well-defined landscape over the parameter space. The classical optimizer navigates that landscape by computing (or estimating) the gradient $\nabla_{\boldsymbol{\theta}} C$ .

A concrete one-parameter example

Consider the simplest non-trivial case: one qubit, one parameter. Apply $R_y(\theta)$ to $|0\rangle$ :

|\psi(\theta)\rangle = R_y(\theta)|0\rangle = \cos\!\tfrac{\theta}{2}|0\rangle + \sin\!\tfrac{\theta}{2}|1\rangle.

Measure in the $Z$ basis. The expectation value of $Z = |0\rangle\langle 0| - |1\rangle\langle 1|$ is

\langle Z \rangle(\theta) = \cos^2\!\tfrac{\theta}{2} - \sin^2\!\tfrac{\theta}{2} = \cos\theta.

The gradient with respect to $\theta$ follows directly from calculus:

\frac{d\langle Z \rangle}{d\theta} = -\sin\theta.

At $\theta = \pi/3$ we get $\langle Z \rangle = \cos(\pi/3) = 1/2$ and $d\langle Z\rangle/d\theta = -\sin(\pi/3) = -\sqrt{3}/2 \approx -0.866$ .

The loss landscape

Because every gate angle enters via a sine or cosine, $C(\boldsymbol{\theta})$ is a trigonometric polynomial — smooth and periodic. This guarantees that gradients exist everywhere, which is what makes gradient descent well-defined for variational quantum algorithms.

A common supervised loss is the mean squared error between the expected output and a target label $y$ :

\mathcal{L}(\boldsymbol{\theta}) = \bigl(\langle O \rangle(\boldsymbol{\theta}) - y\bigr)^2.

By the chain rule:

\frac{\partial \mathcal{L}}{\partial \theta_j} = 2\bigl(\langle O \rangle - y\bigr) \cdot \frac{\partial \langle O \rangle}{\partial \theta_j}.

The first factor comes from the loss shape; the second factor is the gradient of the expectation value, which the parameter-shift rule provides.

Gradient descent update

Given a learning rate $\eta$ the parameters are updated as

\theta_j \leftarrow \theta_j - \eta \,\frac{\partial C}{\partial \theta_j}.

Iterating this loop — run circuit, estimate gradient, update angles — is the variational quantum training loop. The outer loop is classical; only the expectation-value evaluations require quantum resources.

Try it

Compute the expectation value $\langle Z \rangle = \cos\theta$ and its gradient $d\langle Z\rangle/d\theta = -\sin\theta$ at $\theta = \pi/3$ . Return an object { E, grad } with both numbers. The grader checks that both values are numerically correct.

Run your code to see the quantum state.