qDRIFT averaged-channel induced trace-norm error

Let $d,m,N$ be positive integers, $B,t\geq0$, $H_1,\ldots,H_m\in\mathbb{C}^{d\times d}$ be Hermitian matrices with $\|H_j\|\leq B$, and $h_1,\ldots,h_m\geq0$ satisfy $\lambda=\sum_{j=1}^{m}h_j>0$. Set $p_j=h_j/\lambda$ and $T=t\lambda$, and define $E(X)=\sum_{j=1}^{m}p_j e^{-i(T/N)H_j}X e^{i(T/N)H_j}$ and $U(X)=e^{-i(T/N)\sum_{j=1}^{m}p_jH_j} X e^{i(T/N)\sum_{j=1}^{m}p_jH_j}$ for $X\in\mathbb{C}^{d\times d}$. Then $$ \|U^N-E^N\|_{1\to1} \leq B^2\left(e^{TB}(2+TB)+2\right)\frac{T^2}{N}, $$ where powers denote repeated composition, $\|M\|=\sup\{\|Mv\|_2:v\in\mathbb{C}^d,\ \|v\|_2\leq1\}$, $\|X\|_1=\operatorname{Tr}\sqrt{X^\dagger X}$, and $\|\Phi\|_{1\to1}=\sup_{\|X\|_1\leq1}\|\Phi(X)\|_1$.