divide the $M^2$ operators into two groups, one commutes with $V_1$ and the other anti-commutes with $V_1$. Specifically, the following $M-1$ operators $$\begin{aligned}
V_{2} & =& P_{0}+P_{1},\nonumber \\
V_3 &= & P_0 -P_1 +2P_2, \nonumber \\
&\cdots &\nonumber \\
V_{M} & =& P_{0}-P_{1}-...-P_{M-2}+(M-1)P_{M-1}\end{aligned}$$ evidently commutes with $V_1$. In addition, other $(M-2)(M-1)$ basis operators, denoted $V_{M+1},V_{M+2},\cdots,V_{M+(M-2)(M-1)}$, also commute with $V_1$. This is the case because we can construct the following $\frac{1}{2}(M-2)(M-1)$ basis operators $$\begin{aligned}
|k\rangle\langle l |+| l\rangle\langle k|\end{aligned}$$ with $0<k<M$ and $k<l<M$. The other $\frac{1}{2}(M-2)(M-1)$ basis operators that commute with $V_1$ are constructed as $$\begin{aligned}
-i|k\rangle\langle l |+i| l\rangle\langle k|,\end{aligned}$$ also with $0<k<M$ and $k<l<M$. All the remaining $2(M-1)$ basis operators are found to anti-commute with ${V}_{1}$. Specifically, they can be written as $$\begin{aligned}
V_{M+(M-1)(M-2)+2l-1} & =& |0\rangle\langle l|+|l\rangle\langle0|; \nonumber \\
V_{M+(M-1)(M-2)+2l} & =& -i|0\rangle\langle l|+i|l\rangle\langle0|,\end{aligned}$$ where $1\leq l\leq M-1$.

The total Hamiltonian for an uncontrolled $M$-level system interacting with a bath can now be written as $$\begin{aligned}
H_{M}&=& H_{0}+H', \nonumber \\
 H_{0}&=&\sum_{j=1}^{M^{2}-2M+2}W_{j}V_{j}, \nonumber \\
H'&=&\sum_{j=M^{2}-2M+3}^{M^{2}}W_{j}V_{j},\end{aligned}$$ where $W_{j}$ are the expansion coefficients that may contain arbitrary bath operators.

With the UDD control sequence described in Eq. (\[Mc\]) tuned on, the unitary evolution operator can be easily investigated using $[V_1,H_{0}]=0$ and 