Optimal arbitrarily accurate composite pulse sequences

Abstract

Implementing a single-qubit unitary is often hampered by imperfect control. Systematic amplitude errors $\ensuremath{\epsilon}$, 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 $\ensuremath{\pi}$ or $2\ensuremath{\pi}$ rotations that suppress such errors to arbitrary order $O({\ensuremath{\epsilon}}^{n})$ on arbitrary initial states. Optimality is demonstrated by proving an $L=O(n)$ 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 nonrecursive approach, in which finding amplitude error correcting sequences can be reduced to solving polynomial equations.