# Improvement of Quantum Approximate Optimization Algorithm for Max-Cut Problems
## 2 Modified QAOA  
In this section, we present the results of the algorithm applied to a value stream network of \(n=10\) nodes. The complete results for other configurations are available in open access in [23]. We start by representing in Figure 2 the value stream network as a graph \(G=\{n,e\}\) of \(n=10\) unlabeled nodes and \(e=13\) edges. Figure 1: QAOA Overview. If the graph coincides with the connectivity of our logical network (either IoT topology or value stream network), the cost function \(C(x)\) coincides with the hamiltonian \(H\) used to generate the state. For a shallow approximation with \(p=1\), the analytical solution for the expectation value is given by  
\[F_{1}(\beta,\gamma)=\left\langle\psi_{1}|H|\psi_{1}\right\rangle \tag{5}\]  
Combining Equations (3) and (4), the Hamiltonian \(H\) makes use of the expectation value to measure the edges individually:  
\[f_{i,k}(\gamma,\beta)=\frac{1}{2}\left\langle\psi_{1}(\gamma,\beta)|(1-Z_{i}Z _{k})|\psi_{1}(\gamma,\beta)\right\rangle \tag{6}\]  
There are two types of edges: those that connect a node with degree one (A), and those that connect a node with degree three (B). For the A-class edges, an example of the encoding of the optimization function between nodes (0) and (1) is given by  
\[2f_{A}=1-\langle+^{1}|U_{01}(\gamma)U_{12}(\gamma)U_{13}(\gamma)X_{0}(\beta)X _{1}(\beta)Z_{0}Z_{1}X_{1}^{\dagger}(\beta)X_{0}^{\dagger}(\beta)U_{01}^{ \dagger}(\gamma)U_{12}^{\dagger}(\gamma)U_{13}^{\dagger}(\gamma)|+^{1}\rangle \tag{7}\]  
and for the B-class edges, the encoding of the optimization function between nodes (1) and (2) is given by  
\[2f_{B}=1-\langle+^{3}|U_{21}(\gamma)U_{24}(\gamma)U_{23}(\gamma)X_{1}(\beta)X _{2}(\beta)Z_{1}Z_{2}X_{1}^{\dagger}(\beta)X_{2}^{\dagger}(\beta)(\gamma)U_{12 }^{\dagger}(\gamma)U_{23}^{\dagger}(\gamma)U_{24}^{\dagger}(\gamma)|+^{3}\rangle \tag{8}\]  
in which \(|+^{n}\rangle=\sum_{x\in\{0,1\}^{n}}\frac{1}{\sqrt{2^{n}}}\left|x\right\rangle\) prepares for an equal superposition state followed by a sequence of parametrized unitary operations. As shown in Equations (7) and (8) these unitary operations are a combination of parametrized Hamiltonian cost \(U_{C}(\gamma)=e^{-i\gamma H_{C}}\) and mixer layers \(U_{M}(\beta)=e^{-i\beta H_{M}}\). The subindexes in the unitary operations indicate the nodes on which the operators act upon. In our case \(n=10\), there are two A-class edges and eleven B-class edges. This yields Equation (9), which is depicted in Figure 3 which shows the periodicity in both parameters and exhibits a highly nonlinear behaviour.Farhi et al. [20] proposed QAOA with the structure represented in Figure 4.  
\[ \begin{split}& F_{1}(\beta,\gamma)=2f_{A}(\beta,\gamma)+11f_{B}( \beta,\gamma)=\left[\sin(4\gamma)\sin(4\beta)+\sin^{2}(2\beta)\sin^{2}(2\gamma )\right]+\\ &\quad+\ \frac{11}{2}\left[1-\sin^{2}(2\beta)\sin^{2}(2\gamma)\cos^{2}(4 \gamma)-\frac{1}{4}\sin(4\beta)\cos(4\gamma)\Big{(}1+\cos^{2}(4\gamma)\Big{)} \right]\end{split}  \tag{9}\]  
Figure 2: Value stream network with \(n=10\) nodes. Such an algorithm starts by preparing the system in superposition with a Hadamard gate on all qubits. Next, a rotation of \(2\gamma\) is applied to each of the edges if both are in state \(|11\rangle\). This conditional rotation has the form given by  
\[C_{p}(2\gamma)= \begin{pmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&e^{-2i\gamma}\end{pmatrix} . \tag{10}\]  
Figure 4: QAOA—[PERSON] et al. [20]. Figure 3: Analytic solution for \(p=1\) and value stream network configuration from Figure 2.  
This allows the algorithm to be applied when both qubits are in state \(\ket{1}\) simultaneously. A quantum phase correction of \(\gamma\) is then applied to each of the nodes joined by each edge. This rotation has the form given by  
\[p(\gamma)= \begin{pmatrix}1&0\\ 0&e^{i\gamma}\end{pmatrix} . \tag{11}\]  
Such configuration allows the previous rotation to be neutralized when the two qubits are in state \(\ket{11}\).