Optimal arbitrarily accurate composite pulse sequences

Abstract

Implementing a single-qubit unitary is often hampered by imperfect control. Systematic amplitude errors ε, 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 π or 2π rotations that suppress such errors to arbitrary order O(ε[superscript 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.