Parameter Optimization

The VQE loop updates circuit parameters until the energy expectation converges. Two ideas make this possible: the parameter-shift rule gives an exact quantum gradient, and a classical optimizer translates that gradient into a new set of angles.

Math you'll use here: derivatives, gradients, and descent. The derivative $dE/d\theta$ is the slope of the energy curve $E(\theta)$ : how fast $E$ changes per unit change in the angle $\theta$ . A positive slope means $E$ grows as $\theta$ grows, so to lower $E$ you step in the opposite direction. With many angles $\boldsymbol{\theta} = (\theta_1, \dots, \theta_p)$ , the gradient is the vector of partial derivatives,

$\nabla_{\boldsymbol{\theta}} E = \left( \frac{\partial E}{\partial \theta_1}, \dots, \frac{\partial E}{\partial \theta_p} \right),$

where $\partial E / \partial \theta_j$ is the slope along $\theta_j$ with the other angles held fixed. The gradient points in the direction of steepest increase, so gradient descent moves the opposite way, $\boldsymbol{\theta} \leftarrow \boldsymbol{\theta} - \eta\,\nabla_{\boldsymbol{\theta}} E$ , shrinking $E$ a little each step ( $\eta > 0$ is the step size). Below, the parameter-shift rule is just an exact way to measure one such slope $\partial E / \partial \theta$ on a quantum device.

The parameter-shift rule

For any gate of the form $G(\theta) = e^{-i\theta P / 2}$ where $P$ is a Pauli operator, the derivative of the energy expectation $E(\theta) = \langle \psi(\theta) | H | \psi(\theta) \rangle$ with respect to $\theta$ is

$\frac{dE}{d\theta} = \frac{E\!\left(\theta + \tfrac{\pi}{2}\right) - E\!\left(\theta - \tfrac{\pi}{2}\right)}{2}.$

This is not a finite-difference approximation — it is an exact identity, because $\sin$ and $\cos$ satisfy this shift relation. In practice you run the circuit twice (with $\theta$ shifted by $+\pi/2$ and $-\pi/2$ ) and take the scaled difference. Each call to the quantum device gives you one sample of the energy, so a full gradient over $p$ parameters costs $2p$ circuit evaluations.

Gradient descent

Once you have the gradient $\nabla_{\boldsymbol{\theta}} E$ , the simplest update rule is gradient descent:

$\boldsymbol{\theta}^{(t+1)} = \boldsymbol{\theta}^{(t)} - \eta \, \nabla_{\boldsymbol{\theta}} E\!\left(\boldsymbol{\theta}^{(t)}\right),$

where $\eta$ is the learning rate (a small positive number). Because $E \geq E_0$ always holds, the energy can only decrease toward the ground-state energy as long as the step size is small enough to avoid overshooting.

More sophisticated optimizers adapt the step size or use curvature estimates. Gradient-based choices include ADAM and L-BFGS. For shallow VQE circuits, COBYLA (a derivative-free method) and SPSA (a stochastic gradient estimator) are also common choices because they are robust to shot noise.

One full update step

Putting it together:

Run the ansatz at $\boldsymbol{\theta}$ .
For each angle $\theta_k$ , run the ansatz at $\theta_k \pm \pi/2$ and compute $\partial E / \partial \theta_k = \bigl[E(\theta_k + \pi/2) - E(\theta_k - \pi/2)\bigr] / 2$ .
Update $\theta_k \leftarrow \theta_k - \eta \cdot \partial E / \partial \theta_k$ .
Repeat until convergence.

For a single-angle ansatz ( $p = 1$ ), step 2 costs only two extra circuit runs and the update reduces to a scalar subtraction.

Try it

In the exercise the quantum device has already been called: E_plus and E_minus are the measured energies at $\theta + \pi/2$ and $\theta - \pi/2$ . Your task is to apply the parameter-shift rule and return the angle after one gradient-descent step.

Run your code to see the quantum state.