Algorithm 4 QAOA solving optimization.

Input:  the problem Hamiltonian $H_C$, the mixing Hamiltonian $H_M$. Output:  the ground state $|{\mathsf{\Psi }}_C\rangle$ of $H_C$.  1:Prepare the quantum register into $|{\mathsf{\Psi }}_0{\rangle =|+\rangle }^{\otimes m}$.  2:Choose the initial parameters $\mathit{\gamma }$, $\mathit{\beta }$. Perform $H_C$ and $H_M$ alternately and obtain $|\mathsf{\Psi }(\mathit{\gamma },\mathit{\beta })\rangle$.  3:Measure the quantum registers and calculate the cost function.  4:Repeat Step 2 and Step 3 several times and calculate the expectation value of the cost function.  5:Pass the expectation value and parameters $(\mathit{\gamma },\mathit{\beta })$ to a classical optimizer. Update the parameters $(\mathit{\gamma },\mathit{\beta })$.  6:Repeat Steps 2–5 until the result meets a fixed threshold and the parameters are updated to $({\mathit{\gamma }}^{\ast },{\mathit{\beta }}^{\ast })$.  7:return $|{\mathsf{\Psi }}_C\rangle =|\mathsf{\Psi }({\mathit{\gamma }}^{\ast },{\mathit{\beta }}^{\ast })\rangle$