## 2 Proposed Fault Feature Extraction Methodology for the Fault Diagnosis of Rolling Bearings
### Demodulation Analysis Based on CQMA
According to the graph from Reference [40], the adaptive size of structural elements is determined to be:  
\[\lambda_{l}=[\mathit{len}(i)], \tag{25}\]  
where \(\lambda\) is the length variation curve of the structural elements that was fitted by len in Reference [37]:  
\[\mathit{len}(i)=0.7d-0.1 dsiz(i). \tag{26}\]  
The above two formulas can ensure that the size of the structural elements is changed within the range of 0.6\(d\) to 0.7\(d\) to realize self-adaptive adjustment. Then, the heights \(\lambda_{h}\) of the structural elements are determined. First, the basic mathematical expression of quantum structural element (QSE) is established in quantum space, and the QSE that contains n qubits is defined as the vector of quantum probability amplitudes \(\left(\omega_{j}^{r_{j}},\omega_{1}^{0}\right)(j=1,2,\cdots,n)\). \[q\mathcal{s}=\Big{[}\left(\omega_{1}^{r_{1}},\omega_{1}^{0}\right)\left( \omega_{2}^{r_{2}},\omega_{2}^{0}\right),\cdots,\left(\omega_{n}^{r_{n}}, \omega_{n}^{0}\right)\Big{]}, \tag{27}\]  
where \(\omega_{j}^{r_{j}}\) represents the probability amplitude of state \(|1\rangle\) in qubit \(j\) when \(|i_{b}>(j)\) and \(\omega_{j}^{0}\) represents the probability amplitude of state \(|0\rangle\) in qubit \(j\) when \(|i_{b}>(j)\). A mapping method of QSE is established for mapping the QSE from the quantum space to the real space and completing the change of \(|i_{b}\rangle\) to the specified SE. The mapping method is described in detail as follows:  
1. The probability of state \(|1\rangle\) is \(\left|\omega_{j}^{r_{j}}\right|^{2}\) in the case of \(|i_{b}>(j)\). After measurement, a definite single form in quantum space (SFQS) is obtained, and the SFQS is mapped to the single form in real space (SFRS). The SFRS corresponds to bit \(g(j)=r_{j}\), where \(r_{j}\) is a quantitative description of the local feature, and the weighted average of the local time domain parameters of the location of the sampling point is used. 2. The probability of state \(|0\rangle\) is \(\left|\omega_{j}^{0}\right|^{2}\) in the case of \(|i_{b}>(j)\). After measurement, a definite SFQS is obtained, and the SFQS is mapped to the SFRS, with SFRS corresponding to bit \(g(j)=0\). The calculation method for the SFRS height is as follows:
3. Height \(r_{j}\) that corresponds to state \(|1\rangle\):When mechanical failure occurs, the impulse response belongs to the signal that must be extracted. \(r_{j}\) is calculated as the weighted average of the time domain parameters. In SFRS, if the information of the impulse response component is added, it is expected to enhance the impulse response signal in the operation. Consider the kurtosis calculation formula at the Kth sampling point \(s(k)(k=1,2,\cdots,l)\) as an example:  
\[K(k)=\frac{1}{5}\sum_{\omega=-2}^{2}\left(\frac{s\omega(k,\omega)-\overline{s \omega}}{\sigma}\right)^{4}. \tag{28}\]  
The local characteristics of the impulse response signal are calculated to the extent possible, where \(s\omega\) is the signal segment that is intercepted by a window of width 5, \(s\omega(k,\omega)=[s(k-2),s(k+\omega),s(k),\cdots,s(k+2)]\), \(\overline{s\omega}\) is the mean value of \(s\omega(k,\omega)\), and \(\sigma\) is the standard deviation of \(s\omega\). When the CMF operator is applied to the kth sampling point, in combination with the CMF formula, the data segment that operates with SFRS is \([s(k-n+1),s(k-n+2),\cdots,s(k)]\). When SFQS is mapped to SFRS, \(r_{j}\) in the \(j\) th position is as follows:  
\[r_{j}=K(k-j+1)(j=1,2,\cdots,n). \tag{29}\]  
When \(|I_{b}\rangle=|11\cdots 1\rangle\), \(g=[r_{1},r_{2},\cdots,r_{n}]\), the SFRS is completely determined by the weighted average of the time domain parameters. Height that corresponds to state \(|0\rangle\):  
When \(|I\rangle=|00\cdots 0\rangle\), the height of SFQS is 0 for SE--namely, \(g=[0,0,\cdots,0]\)--and SE is degraded into a flat SE with a height of zero. After mapping, QSE will generate a different SFRS, and the quantum probabilities will differ between the SFRSs.