to $$\begin{aligned}
U(T) & = & (-iY_1)^{N}\mathfrak{J}\left[ e^{-i\int _{T_N}^{T}[H_{0}+(-1)^{N} H'] dt}\right]  \nonumber \\
& &\ \times\ \mathfrak{J}\left[e^{-i\int_{T_{N-1}}^{T_N}[H_{0}+(-1)^{N-1}H']dt} \right] \nonumber \\
& &\ \cdots \nonumber \\
 & &\ \times\ \mathfrak{J}\left[ e^{-i\int_{T_2}^{T_3}[H_{0}+H']dt} \right]\nonumber \\
& &\ \times\ \mathfrak{J}\left[ e^{-i\int_{T1}^{T2} [H_{0}-H']dt} \right]\nonumber \\
& &\ \times\  \mathfrak{J}\left[e^{-i\int_{0}^{T_1}[H_{0}+H']dt}\right] \nonumber \\
&=& (-iY_1)^{N}\mathfrak{J} \left[e^{-i\int_{0}^{T}H_0 dt}\right] \mathfrak{J}\left[ e^{-i\int_{0}^{T}
F_{N}(t)H_{I}'(t)\mathrm{d}t}\right], \nonumber \\
\label{uorder}\end{aligned}$$ with $$\begin{aligned}
H_I'(t) = \mathfrak{J} \left[e^{i\int_{0}^{T}H_0 dt}\right] H' \mathfrak{J} \left[e^{-i\int_{0}^{T}H_0 dt}\right].\end{aligned}$$ Because the term $\mathfrak{J} \left[e^{-i\int_{0}^{T}H_0 dt}\right]$ in Eq. (\[uorder\]) does not affect the expectation value of our coherence measure, the final expression for the coherence measure is essentially the same as before and is hence again given by its initial value multiplied by $1+O(T^{N+1})$.

Our construction of the UDD control sequence is based on a pre-determined coherence measure ${\cal P}_{|\Psi\rangle}$ that characterizes a certain type of quantum coherence. This implies that our two-qubit UDD relies on which type of decoherence we wish to suppress. Indeed, this is a feature shared by Uhrig’s work [@UDDprl] and the Yang-Liu universality proof [@univUDD] for single-qubit systems (i.e., suppressing either transverse decoherence or longitudinal population relaxation). Can we also efficiently suppress decoherence of different types at the same time, or can we simultaneously preserve the quantum coherence 