High-order discrete adiabatic convergence under boundary cancellation

Let $W(s)$) be a discrete evolution operator whose eigenvalues are separated into groups $\sigma_P(s)$ and $\sigma_Q(s)$ as in the stated adiabatic assumptions, let $\left\vert\psi\right\rangle$ lie in the eigenspace corresponding to $\sigma_P(0)$, and let $P(s)$ denote the spectral projection of $W(s)$ onto $\sigma_P(s)$. Suppose that $W(s)$ satisfies the boundary cancellation condition

$$W^{(j)}(0)=W^{(j)}(1)=0\quad\text{for all }j\geq 1.$$

For any integer $T$ and any positive integer $k$, the discrete evolution $U(s)$ satisfies

High-order discrete adiabatic convergence conjecture.

$$\left\|U(1)\left\vert\psi\right\rangle-P(1)U(1)\left\vert\psi\right\rangle\right\|\leq\frac{C_k}{T^k},$$

where $C_k$ is independent of $T$. The conjecture asserts super-polynomial suppression of the discrete diabatic error under boundary cancellation. The cited proof has a missing step, so the result is not rigorously established, although numerical tests support the claimed convergence.