## 2 Proposed Fault Feature Extraction Methodology for the Fault Diagnosis of Rolling Bearings
### Demodulation Analysis Based on CQMA  
Due to the lack of prior information on the impact components, it is difficult to select structural elements and their lengths for the fault signals that are collected on-site. Each signal has local characteristics that change with time and lacks the detailed processing ability of fixed structural element length, and it is impossible to realize the optimal processing of the signal [40]. Therefore, we propose a mathematical morphological analysis method that is based on quantum theory optimization and can adaptively adjust the sizes of structural elements according to the local characteristics of the signals. According to the literature [35], when analyzing and processing bearing vibration signals, the most commonly used and effective structural unit is the triangular structural unit, which can extract the fault characteristic information of the bearing well. Therefore, this paper selects triangular structural elements for morphological analysis so that the sizes of the structural elements (the length and height of the triangle) can be adjusted adaptively according to each signal. Since the key component of the vibration signal quantum system is the vibration signal qubit, this section presents the mathematical expression of the vibration signal qubit. Assume that the sensor acquisition signal is \(s(i)(i=0,1,2,\cdots,n-1)\) and normalize \(s(i)\) to obtain signal \(y(i)\in[0,1]\). The adaptive threshold \(T\in(s_{\min},s_{\max})\) is introduced, and the normalization transformation is as follows:  
\[y(i)=\left\{ \begin{array}{c}\frac{s(i)-s_{\min}}{2(T-s_{\min})},s(i)\leq T \\ 0.5+\frac{s(i)-T}{2(s_{\max}-T)},\text{others}\end{array} \right.. \tag{19}\]  
Although the vibration signal contains nonstationary and nonlinear components and various interference noises, it is statistical in essence. According to the processing of information from the perspective of probability statistics, a mathematical expression for the vibration signal quantum bit is proposed in Formula (20), which is used to realize the mapping of the vibration signal from time domain space to quantum space. \[\left|y_{0}(i)\right.\rangle=\sqrt{1-y(i)}\left|0\right\rangle+\sqrt{y(i)}\left| 1\right\rangle, \tag{20}\]  
where \(\left|0\right\rangle\) and \(\left|1\right\rangle\) are the two ground states in the vibration signal quantum bit, which correspond to the minimum \(s(i)_{\text{min}}\) state and the maximum \(s(i)_{\text{max}}\) state of the vibration signal; \(\sqrt{1-y(i)}\) and \(\sqrt{y(i)}\) are the probability amplitudes of the two ground states; and \(1-y(i)\) and \(y(i)\) represent the probabilities of occurrence of the minimum \(s(i)_{\text{min}}\) and maximum \(s(i)_{\text{max}}\), respectively, in the vibration signal. Based on the vibration signal quantum bit, the length \(\lambda_{l}\) of the structural elements is defined. Combined with the correlation of vibration signals, the length measurement operator (LMO) of structural elements of mechanical vibration signals under quantum probability characteristics is proposed to guide the adaptive selection of structural element size. The neighboring window can form a 3-qubit system, and its state vector is \(\left|s(i-1)s(i)s(i+1)\right\rangle\). The normalized value \(y(i)\in[0,1]\) of \(s(i)\) is obtained via Formula (20).