Arbitrarily accurate variable rotations on the Bloch sphere by composite pulse sequences

TL;DR

This paper introduces composite pulse sequences capable of achieving arbitrary rotations on the Bloch sphere with arbitrary accuracy, compensating for experimental errors and enabling precise control of two-state quantum systems.

Contribution

It presents a method to construct arbitrarily accurate composite pulse sequences for arbitrary rotations, including special classes of sequences with simple phase formulas for $ heta/2$ rotations.

Findings

Composite sequences can compensate errors to any order.

Explicit formulas for phases of symmetric and asymmetric $ heta/2$ sequences.

Construction of long composite sequences from $ heta/2$ sequences.

Abstract

Composite pulse sequences, which produce arbitrary pre-defined rotations of a two-state system at an angle $\theta$ on the Bloch sphere, are presented. The composite sequences can contain arbitrarily many pulses and can compensate experimental errors in the pulse amplitude and duration to any desired order. A special attention is devoted to two classes of $\pi/2$ sequences --- symmetric and asymmetric --- the phases of which are given by simple formulas in terms of rational multiples of $\pi$ for any number of constituent pulses. This allows one to construct arbitrarily accurate $\pi/2$ composite rotations. These $\pi/2$ composite sequences are used to construct three classes of arbitrarily long composite $\theta$ sequences by pairing two $\pi/2$ composite sequences, one of which is shifted by a phase $\pi-\theta$ with respect to the other one.