## 3 Hierarchical Optimization Algorithm
### Optimization Process  
Owing to the multiple objectives and parameters involved in the optimization solution, a hierarchical optimization approach is used for this optimization model based on computational efficiency considerations when selecting\(f_{\text{USO}}\),\(f_{\text{PT}}\),\(f_{\text{ADC}}\),\(f_{\text{upper}}\),\(f_{\text{lower}}\), and \(m\). The optimization process is as follows. Step 1: A multi-objective optimization algorithm is used to set the lower and upper bounds of the beat-note frequency (\(f_{\text{lower}},f_{\text{upper}}\)), USO frequency \(f_{\text{USO}}\), and sideband modulation depth \(m\). The multi-objective optimization model for this step is expressed by Equation (10). \[\left\{ \begin{array}{l}\Gamma_{1}(f_{\text{upper}},f_{\text{USO}},\overline{ f_{\text{PT}}},\overline{f_{\text{ADC}}},m,\overline{P_{\text{local}}})&=\min \bigl{(}\delta x^{\text{total}}\bigr{)}\\ \Gamma_{2}(f_{\text{lower}})&=\max(f_{\text{lower}})\\ \Gamma_{3}(f_{\text{upper}})&=\min(f_{\text{upper}})\\ \Gamma_{4}(f_{\text{upper}},f_{\text{lower}})&=\max(f_{\text{upper}}-f_{\text{ lower}})\\ s.t.\\ \left\{\begin{array}{l}0.44<m<0.61\\ f_{\text{USO}}<5\,\text{GHz}\\ 2\;\text{MHz}<f_{\text{lower}}<10\;\text{MHz}\\ 20\;\text{MHz}<f_{\text{upper}}<=25\;\text{MHz}\end{array}\right.\end{array}  \right.\text{.} \tag{10}\]  
In Equation (10), \(\Gamma_{x}\) denotes the x-th optimization objective. In \(\Gamma_{1}\), where \(\overline{f_{\text{PT}}}\), \(\overline{f_{\text{ADC}}}\) and \(\overline{P_{\text{local}}}\) denote that the variables are taken as constant. \(\Gamma_{2}\) and \(\Gamma_{3}\) in Equation (10) are the two objective functions derived from \(\Gamma_{1}\) in Equation (9) according to Equations (3) and (8), respectively. \(\Gamma_{2}\) aims to expand the selection spaces of \(f_{\text{ADC}}\) and \(f_{\text{PT}}\), and \(\Gamma_{3}\) aims to reduce the sideband readout displacement noise, which is proportional to \(f_{\text{upper}}\). The objective function \(\Gamma_{4}\) aims to maximize the feasible range of the beat-note frequency, which is the opposite of \(\Gamma_{2}\) and \(\Gamma_{3}\). In actual mission operations,\(f_{\text{upper}}\),\(f_{\text{lower}}\), and \(f_{\text{USO}}\) are commonly set as integers. To reduce the complexity of the optimization problem while determining the solution,\(f_{\text{upper}}\),\(f_{\text{lower}}\), and\(f_{\text{USO}}\) are not constrained as integers in this step. In the current Taiji program, the upper bound of the beat-note frequency is 25 MHz. Using the parameter values listed in Table 1, \(P_{\text{receive}}\) = 2.154 \(\times\) 10\({}^{-9}\) W. Therefore, the values of all parameters in Equation (4), except \(P_{\text{local}}\), are obtained. Because \(P_{\text{local}}\) appears in both the numerator and denominator, the optimal value of \(P_{\text{local}}\), which is 2.06 \(\times\) 10\({}^{-3}\) W in the current parameter settings, can be derived by simply minimizing. When calculating \(\delta x^{\text{total}}\) in this step, the values of \(P_{\text{local}}\) = 2.06 \(\times\) 10\({}^{-3}\) W and the parameters in Table 1 are used by default. Step 2: The values of \(f_{\text{lower}}\), \(f_{\text{upper}}\), and \(f_{\text{USO}}\) obtained in Step 1 are adjusted to be integers:  
\[\left\{ \begin{array}{l}f_{\text{lower}}^{\text{new}}=\text{ floor}(f_{\text{lower}})\\ f_{\text{upper}}^{\text{new}}=\text{ floor}(f_{\text{upper}})\\ f_{\text{USO}}^{\text{new}}=\text{ floor}(f_{\text{USO}})\end{array} \right., \tag{11}\]  
where \(f_{\text{lower}}^{\text{new}}\), \(f_{\text{upper}}^{\text{new}}\), and \(f_{\text{USO}}^{\text{new}}\) represent the adjusted values of \(f_{\text{lower}}\), \(f_{\text{upper}}\), and \(f_{\text{USO}}\), respectively. Step 3: Exhaustive enumeration is used to search for possible \([f_{\text{USO}},f_{\text{PT}},f_{\text{ADC}}]\) combinations. The values of \(f_{\text{PT}}\) and \(f_{\text{ADC}}\) need to satisfy the following constraints:  
\[\left\{