Parameterizing Layers

A single QAOA layer is a rough approximation. Stacking $p$ layers — each with its own pair of angles $(\gamma_k, \beta_k)$ — gives the optimizer more freedom and generally produces better solutions. This lesson builds the $p = 2$ circuit from scratch.

QAOA recap: one layer

For a problem graph with edges $E$, the cost Hamiltonian is

A single QAOA layer applies two unitaries in sequence:

where $H_B = \sum_i X_i$ is the mixing Hamiltonian. Starting from the uniform superposition $|s\rangle = H^{\otimes n}|0\rangle^{\otimes n}$, one full layer reads

Going deeper: $p$ layers

With $p$ layers the ansatz becomes

The $2p$ real parameters $(\gamma_1,\ldots,\gamma_p,\beta_1,\ldots,\beta_p)$ are tuned by a classical optimizer to maximize $\langle H_C\rangle$. As $p\to\infty$, the QAOA state can exactly represent the true optimum; finite $p$ trades circuit depth for solution quality.

Building the circuit gates

For a graph with a single edge $(0, 1)$, the ZZ interaction $e^{-i\gamma Z_0 Z_1/2}$ is compiled as three gates:

The factor of $1/2$ in the Hamiltonian means the angle passed to $R_Z$ is $\gamma$ (one factor of 2 from the eigenvalue of $Z_0 Z_1$, one from the exponent convention of $R_Z$).

The mixer $e^{-i\beta X_q}$ on each qubit $q$ is a single rotation gate:

The extra factor of 2 arises because $R_X(\theta) = e^{-i(\theta/2)X}$.

Structure of the $p = 2$ circuit

q0: ─H─[cx]─────────[cx]─Rx(2β₁)─[cx]─────────[cx]─Rx(2β₂)─
q1: ─H─[cx]─Rz(γ₁)─[cx]─Rx(2β₁)─[cx]─Rz(γ₂)─[cx]─Rx(2β₂)─

The superposition is prepared once; then each layer alternates the cost and mixer unitaries with independent angles.

Try it

Complete the $p = 2$ circuit using the fixed angles $\gamma_1 = 0.4$, $\gamma_2 = 0.8$, $\beta_1 = 0.3$, $\beta_2 = 0.6$. The grader compares the full statevector, so every gate placement matters.

Sign in on the full site to ask questions and join the discussion.