  Implementing a single qubit unitary is often hampered by imperfect control.
Systematic amplitude errors $\epsilon$, caused by incorrect duration or
strength of a pulse, are an especially common problem. But a sequence of
imperfect pulses can provide a better implementation of a desired operation, as
compared to a single primitive pulse. We find optimal pulse sequences
consisting of $L$ primitive $\pi$ or $2\pi$ rotations that suppress such errors
to arbitrary order $\mathcal{O}(\epsilon^{n})$ on arbitrary initial states.
Optimality is demonstrated by proving an $L=\mathcal{O}(n)$ lower bound and
saturating it with $L=2n$ solutions. Closed-form solutions for arbitrary
rotation angles are given for $n=1,2,3,4$. Perturbative solutions for any $n$
are proven for small angles, while arbitrary angle solutions are obtained by
analytic continuation up to $n=12$. The derivation proceeds by a novel
algebraic and non-recursive approach, in which finding amplitude error
correcting sequences can be reduced to solving polynomial equations.
