Let's say that I have a curve in $\mathbb{C}^N$ given by the action of the unitary group: $$x(t) = e^{Ht}x_0,~ H \in \mathfrak{u}(N),~ ||x_0||=1$$ Here, $H$ is an NxN skew-Hermitian matrix (for very large $N$). I can approximate this to first order as: $$\tilde x(t) \approx x_0 + t(Hx_0) + O(t^2)$$ However, this map isn't unitary; $||\tilde x(t)|| \neq 1$. A better first-order approximation seems to be a great circle on the unit sphere in $\mathbb{C}^N$: $$\tilde x(t)' \approx \cos(\alpha t)x_0 + \alpha^{-1}\sin(\alpha t)Hx_0,~\alpha = ||Hx_0||$$ My question is: Is there a simple generalization of this to higher order terms? Specifically, I'm looking for a family of curves: $$\gamma_M(t) : \mathbb{R} \rightarrow S_C$$ that satisfy: $$\frac{d^k}{dt^k} \gamma_M(t)|_{t=0} = \begin{cases}(H^k)x_0 & (k \le M) \\ 0 & (k > M)\end{cases}$$ where $S_C$ is the unit sphere in $\mathbb{C}^N$. It should be obvious that: $$\lim_{M\rightarrow\infty} \gamma_M(t) = \exp(Ht)x_0$$

For context, $H$ represents a complicated linear combination of elements of $\mathfrak{u}(N)$, and is intractable to compute explicitly (thus making the true exponential map nearly impossible to compute). However, I can compute the action of $H$ on vectors, so I want the closest easy-to-compute unitary transform as a function of some number of iterated applications of $H$.