Classical Optimization of Angles

QAOA circuits contain free parameters — the angle vectors $\vec\gamma$ and $\vec\beta$ — that control how well the quantum state concentrates on high-quality solutions. Picking good parameters is not a quantum task; it is a classical optimization problem that sits in a feedback loop around the quantum circuit.

The expectation-value landscape

After running the QAOA circuit and measuring, you can estimate

\langle C(\vec\gamma,\vec\beta)\rangle = \langle\psi(\vec\gamma,\vec\beta)|\,H_C\,|\psi(\vec\gamma,\vec\beta)\rangle,

the average cost over many shots. This number is a smooth function of the $2p$ real parameters, and the classical optimizer's job is to find the angles that maximize it.

For $p = 1$ Max-Cut on a single edge $(q_0, q_1)$ the expectation value has a closed form. Starting from $|{+}{+}\rangle = H^{\otimes 2}|00\rangle$ and applying the cost unitary $U_C(\gamma)$ followed by the mixer $U_B(\beta)$ gives a state whose overlap with the cut subspace can be computed analytically. The result is

\langle C(\gamma,\beta)\rangle = \tfrac{1}{2} + \tfrac{1}{4}\sin(2\gamma)\sin(2\beta).

You can verify this by expanding the state in the Bell basis or by direct matrix multiplication; the derivation follows Appendix B of Farhi et al. (2014).

Finding the optimal angles

The formula shows that $\langle C \rangle$ is maximised when both sine factors equal $+1$ :

\sin(2\gamma^*) = 1 \implies 2\gamma^* = \tfrac{\pi}{2} \implies \gamma^* = \tfrac{\pi}{4},

\sin(2\beta^*) = 1 \implies \beta^* = \tfrac{\pi}{4}.

Substituting back:

\langle C \rangle_{\max} = \tfrac{1}{2} + \tfrac{1}{4}\cdot 1 \cdot 1 = \tfrac{3}{4}.

The approximation ratio is $\langle C \rangle_{\max} / C_{\max} = (3/4) / 1 = 3/4$ , confirming that $p = 1$ QAOA beats the trivial random-guess bound of $1/2$ for this instance.

Classical optimizers in practice

In production QAOA implementations the loop looks like this:

Propose a candidate $(\gamma, \beta)$ — either analytically (as above for $p = 1$ ) or via a gradient-free method.
Execute the parameterized circuit on a quantum backend (or simulator).
Estimate $\langle C \rangle$ by averaging the shot outcomes.
Update the parameters using the optimizer's rule.
Repeat until convergence.

The quantum computer is treated as a black-box function evaluator; the intelligence about which angles to try lives entirely in the classical optimizer.

For real devices, noise and shot sampling add stochastic error to each $\langle C \rangle$ estimate, so the optimizer must be robust to noisy gradients. Gradient-based methods can use the parameter-shift rule:

\frac{\partial \langle C \rangle}{\partial \gamma_k} = \langle C \rangle\!\big|_{\gamma_k + \pi/4} - \langle C \rangle\!\big|_{\gamma_k - \pi/4}.

This exact identity holds because each QAOA cost layer applies $e^{-i\gamma_k Z_j Z_k}$ (generator eigenvalues $\pm 1$ , shift $= \pi/(4\cdot 1) = \pi/4$ , coefficient $= 1$ ). It requires two extra circuit evaluations per parameter but avoids finite-difference noise amplification.

Try it

Use the closed-form formula to compute the maximum achievable $\langle C \rangle$ for $p = 1$ single-edge Max-Cut, and return it. No circuit needed — this is a purely classical calculation.

Run your code to see the quantum state.