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 $H$.