numerical experiments. Section IV discusses the implications of our results and then concludes this paper.

UDD Theory for two-qubit and general multi-level systems 
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On Yang-Liu’s Universality Proof for Single-Qubit Systems
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For our later use we first briefly describe one central result in Yang and Liu’s work [@univUDD] for proving the universality of the UDD control sequence applied to single-qubit systems. Let $C$ and $Z$ be two time-independent Hermitian operators. Define two unitary operator $U^{(N)}_{\pm}$ as follows: $$\begin{aligned}
\label{oldun}
 U^{(N)}_{\pm}(T)& =& e^{-i[C\pm (-1)^N Z](T-T_N)} \nonumber \\
&&\times\   e^{-i[C\pm (-1)^{(N-1)} Z](T_N-T_{N-1})} \cdots  \nonumber \\
           &&\times\ e^{-i[C\mp Z](T_2-T_1)} e^{-i[C\pm Z]T_1}
           .
$$ Yang and Liu proved that for $T_j$ satisfying Eq. (\[Tj\]), we must have $$\left(U^{(N)}_{-}\right)^{\dagger} U^{(N)}_{+} = 1+O(T^{N+1}),
\label{Tn1}$$ i.e., the product of $\left(U^{(N)}_{-}\right)^{\dagger}$ and $U^{(N)}_{+}$ differs from unity only by the order of $O(T^{N+1})$ for sufficiently small $T$. In the interaction representation, $$\begin{aligned}
Z_{I}(t) & \equiv & e^{iCt}Ze^{-iCt} \nonumber \\
&=& \sum_{p=0}^{\infty}\frac{(it)^{p}}{p!}\underbrace{[C,[C,...[C,Z]]]}_{p\ \mathrm{folds}},\end{aligned}$$ hence the above expression for $U^{(N)}_{\pm}$ can be rewritten in the following compact form $$\begin{aligned}
\label{upm}
U^{(N)}_{\pm}(T)=e^{-iCT}\mathfrak{J}\left[ e^{-i\int_{0}^{T}\pm F_{N}(t)Z_{I}(t)\mathrm{d}t}\right],\end{aligned}$$ where $T$ is the final time, $\mathfrak{J}$ denotes the time-ordering operator, and $$\begin{aligned}
 F_{N}(t)=(-1)^j,\ {\rm for}\ \ t\in (T_j, T_{j+1}).
 \label{fn}
 \end{aligned}$$ As an important observation, we 