## 3 HSP-QGA Algorithm Design
### Algorithm Design
#### 3.2.3 Full-Interference Crossover  
The crossover operator in the traditional genetic algorithm is limited to two individuals. When two individuals are the same, the crossover operation no longer works, and the population diversity decreases. In this paper, a new method: full-interference crossover is designed using quantum coherence. By crossover of each chromosome (processor serial number) in the population, chromosome variation is increased, and results are prevented from falling into the local optimum. A simplified example is as follows:  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Method of Operation** & \multicolumn{8}{c}{**Method of Encoding**} \\ \hline Qubit encoding & \(\begin{pmatrix}\alpha_{11}&\alpha_{12}&\alpha_{21}&\alpha_{22}&\alpha_{31}& \alpha_{32}&\alpha_{41}&\alpha_{42}&\alpha_{51}&\alpha_{52}\\ \beta_{11}&\beta_{12}&\beta_{21}&\beta_{22}&\beta_{31}&\beta_{32}&\beta_{41} &\beta_{42}&\beta_{51}&\beta_{52}\end{  
There is a population of five individuals, each with five chromosomes, where I (1)-I (5) represents the first individual (a processor scheduling strategy), I (1) represents the first chromosome (processor serial number) in the individual, and other individual manifestations are sub-analogous. The full-interference crossover operation of the population is shown in Table 3. The first chromosome of each individual remains unchanged. Starting from the second chromosome, the corresponding chromosomes of all individuals turn downwards to achieve full-interference crossover operation.  
To illustrate and compare the role of the full interference crossover operation in the overall algorithm, we visualize the output of the scheduling length for each step in the algorithm. A scatter plot of the solutions after the full-interference crossover operation in each generation is shown in Figure 5. The full-interference crossover operation increases the diversity of the population and provides a sufficiently large solution space at the beginning of the iteration. A scatter plot of the nodes of the full flow of the algorithm is shown in Figure 6. Based on the solution space provided by the full- interference crossover operation, the algorithm accelerates the mutation (evolution) of the population toward the optimal solution using a quantum rotation gate. The algorithm converges after the 25 th iteration. The full-interference crossover and variance converge to stable values and no longer play a role in the algorithm.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Example**} & \multirow{2}{*}{**Individual**} & \multicolumn{5}{c}{**Chromosome**} \\ \cline{3-6}  & & **First** & **Second** & **Third** & **Fourth** & **Fifth** \\ \hline \multirow{4}{*}{Before full interference crossover} & 1 & I (1) & I (2) & I (3) & I (4) & I (5) \\  & 2 & II (1) & II (2) & II (3) & II (4) & II (5) \\  & 3 & III (1) & III (2) & III (3) & III (4) & III (5) \\  & 4 & IV (1) & III (2) & IV (3) & IV (4) & IV (5) \\  & 5 & V (1) & V (2) & V (3) & V (4) & V (5) \\ \hline \multirow{4}{*}{After full interference crossover} & 1 & I (1) & V (2) & IV (3) & III (4) & II (5) \\  & 2 & II (1) & I (2) & V (3) & IV (4) & III (5) \\ \cline{1-1}  & 3 & III (1) & II (2) & I (3) & V (4) & IV (5) \\ \cline{1-1}  & 4 & IV (1) & III (2) & II (3) & I (4) & V (5) \\ \cline{1-1}  & 5 & V (1) & IV (2) & III (3) & II (4) & I (5) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Example of full-interference crossover operation.  
Figure 5: Scatter plot of the solution after full interference crossover operation.