## 2 Proposed Fault Feature Extraction Methodology for the Fault Diagnosis of Rolling Bearings
### Demodulation Analysis Based on CQMA
The quantum probability amplitude \(\left(\omega_{j}^{r_{j}},\omega_{1}^{0}\right)\) is related to the probabilities of SFRSs, and the morphological results depend on the SFRSs; hence, it is necessary to set the probability amplitude reasonably, and the probability setting depends mainly on the randomness of the signal. Additional restrictions are required, and \(\left(\omega_{j}^{r_{j}},\omega_{1}^{0}\right)\) must satisfy the following two conditions:  
\[ \begin{array}{l}0\leq\omega_{j}^{r_{j}}\leq 1\\ 0\leq\omega_{j}^{0}\leq 1\end{array} . \tag{30}\]  
Since the normalized vibration signal satisfies the above equations, it can be directly used to represent the probability amplitude of the ground state. The sampled signal \(s(k)\) is normalized via Formula (30). When the kth sampling point is processed by the CMF operator, the data segment that calculated with the SFRS is \([s(k-n+1),s(k-n+2),\cdots,s(k)]\), and the ground state probability amplitude \(|1\rangle\), which is expressed by the normalization result of the vibration signal in the case of \(|I_{b}\rangle>(j)\), is as follows:  
\[\omega_{j}^{r_{j}}=\sqrt{z(k-j+1)}. \tag{31}\]  
To satisfy the normalization condition of the quantum system, the probability amplitude of ground state \(|0\rangle\) in the case of \(|I_{b}\rangle>(j)\) is determined to be:  
\[\omega_{j}^{0}=\sqrt{1-z(k-j+1)}. \tag{32}\]  
The expansion form of QSE (EFQS) in quantum space is as follows:  
\[ \begin{array}{l}\left|qs\_ex=\left(\omega_{0}^{1}\times\omega_{0}^{2}\times \cdots\times\omega_{0}^{j}\times\cdots\times\omega_{0}^{n}\right)|00\cdots 0 >+\right.\\ \left(\omega_{0}^{1}\times\omega_{0}^{2}\times\cdots\times\omega_{0}^{j}\times \cdots\times\omega_{1}^{n}\right)|00\cdots 1>+\\ \left.\cdots+\left(\omega_{1}^{1}\times\omega_{1}^{2}\times\cdots\times\omega_ {1}^{j}\times\cdots\times\omega_{1}^{n}\right)|11\cdots 1>\end{array} . \tag{33}\]After quantum measurement, the above equation collapses to a ground state--namely, SFQS--and the expression is as follows:  
\[|i_{b}>\in\{|00\cdots 0>,\cdots,|11\cdots 1>\}. \tag{34}\]  
The probability amplitude \(W_{i}\) of \(|i_{b}>\) is as follows:  
\[W_{i}=\Big{\{}\omega_{0}^{1}\times\omega_{0}^{2}\times\cdots\times\omega_{0}^{n },\cdots,\omega_{1}^{1}\times\omega_{1}^{2}\times\cdots\times\omega_{1}^{n} \Big{\}}. \tag{35}\]  
Combined with the mapping method, the formula \(|i_{b}>\in\{|00\cdots 0>,\cdots,|11\cdots 1>\}\) is mapped to the real space, the weighted average of time domain parameters and 0 are selected as the height with different probabilities, SFQS is changed to SFRS, and the expression is as follows:  
\[SFRS\in\{[0,0,\cdots,0],[0,0,\cdots,r_{n}],\cdots,[r_{1},r_{2},\cdots,r_{n}]\}. \tag{36}\]  
This corresponds to the square of the probability amplitude of the SFQS. Therefore, the composite formula in which CQSE is SFRS is as follows:  
\[\lambda_{h}=|W_{0}|^{2}[0,0,\cdots,0]+|W_{1}|^{2}[0,0,\cdots,r_{n}]+ \tag{37}\] \[\cdots+|W_{2n-1}|^{2}[r_{1},r_{2},\cdots,r_{n}]\]  
Finally, the size of the selected structural elements is demodulated and analyzed using the morphological operator. In this paper, we use a combined morphological filter (CMF) as the morphological operator. In summary, the calculation steps of the CQMA method that is proposed in this paper are as follows:  
Step 1: Read in the reconstructed vibration signal. Step 2: The initialization parameter is set to \(T=s_{\text{min}}+0.1\) and the variables to opt_l = 0, opt_h = 0, opy_T = 0, where opt_h and opt_T are used to store the current maximum information entropy and its corresponding threshold T, respectively. Step 3: Incorporate the threshold parameter into the normalization formula to obtain \(y(i)\). Step 4: For each sampling point, calculate the dimensions \(\lambda_{l},\lambda_{h}\) of the CQMA structural elements and construct a combined morphological analyzer. Step 5: According to each sampling point, a combined morphological analyzer is used for analysis to obtain the vibration signal. Step 6: Calculate the information entropy of the final signal.