Optimal arbitrarily accurate composite pulse sequences

TL;DR

This paper develops optimal composite pulse sequences that correct systematic amplitude errors in single qubit operations to arbitrary order, providing explicit solutions and a novel algebraic approach for their construction.

Contribution

It introduces a new algebraic, non-recursive method to design optimal composite pulses that suppress amplitude errors to any desired order, with explicit solutions for multiple error suppression levels.

Findings

Sequences of length 2n achieve error suppression to order n.

Closed-form solutions are provided for n=1,2,3,4.

Perturbative solutions are proven for small angles, and analytic continuation extends solutions up to n=12.

Abstract

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.