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

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.