# Fault Diagnosis in Partially Observable Petri Nets with Quantum Bayesian Learning
## 4 Case Analysis and Verification
### Algorithm Comparison  
We use the BPN algorithm and the QBPN algorithm, respectively, to diagnose the same system state. After \(100\) simulations, we obtain the comparison diagram of the BPN algorithm and the QBPN algorithm, as shown in Figure 12. In Figure 13, the probability distribution interval of the BPN algorithm is \([0.9425,0.975]\), which is \(0.961\) after the curve fitting. In Figure 14, the probability distribution interval of the QBPN algorithm is \([0.9525,0.980]\), which is \(0.966\) after the curve fitting.  
It can be concluded from Figure 12 that the QBPN algorithm can obtain better probability estimation after quantum interference correction.  
In Table 5, we compare the performance of the BPN algorithm and the QBPN algorithm in this paper. On the basis of the case model, we diagnose the observable fault transitions by using the BPN algorithm. For the most complex OLPS considerable label probability sequence: after many tests, the average response time of the BPN algorithm is \(300\) ms. By applying the QBPN algorithm in this paper, the average response time of fault diagnosis based on the subnet model established for fault transitions is \(230\) ms. It is proved that the QBPN algorithm reduces the computational complexity and improves the efficiency. Meanwhile, in \(5000\) simulation experiments, the number of correct diagnoses the BPN algorithm is \(4685\). The accuracy of diagnosis is \(93.70\%\). The QBPN algorithm modifies  
Figure 12: Comparison of fault diagnosis probability between the BPN algorithm and the QBPN algorithm.  
3. Based on the POPN fault diagnosis model, the QBPN model is simulated and verified through the simulation platform, which verifies the reliability of the QBPN algorithm and can reduce the complexity of the algorithm.  
The complexity of the QBPN algorithm based on the POPN model depends on the number of possible fault transitions, and needed plenty of time to analyze the probabilistic data in the QBPN subnet. We studied the POPN model of transition generalized constraint conditions to reduce the search path of the fault transition set to achieve the purpose of optimizing the algorithm rate. In this paper, we executed the parameter selection of the quantum Bayesian interference term manually without randomness. In the following part, we will study the influence on the parameter selection of the quantum interference term, and attempt to obtain a more realistic transition probability distribution through the MCMC random sampling algorithm to further improve the accuracy of the fault probability of the system.  
Conceptualization, J.L. and E.S.B.M.; methodology, J.L.; software, J.L.; validation, J.L., X.Z. and A.D.; formal analysis, X.Z.; investigation, A.D.; resources, A.D.; data curation, J.L.; writing--original draft preparation, J.L.; writing--review and editing, X.Z.; visualization, E.S.B.M.; supervision, X.Z. All authors have read and agreed to the published version of the manuscript.  
The research has been partially supported by the National Natural Science Funds (61473144).  
Not applicable.  
The data presented in this study are available in article.  
The authors declare no conflict of interest.