  Composite pulse sequences designed for nuclear magnetic resonance experiments
are currently being applied in many quantum information processing
technologies.We present an analysis of a family of composite pulse sequences
used to address systematic pulse-length errors in the execution of quantum
gates. It has been demonstrated by Cummins et al. [Phys. Rev. A 67, 042308
(2003)] that for this family of composite pulse sequences, the fidelity of the
resulting unitary operation compared with the ideal unitary operation is
1+C*epsilon^6, where epsilon is the fractional error in the length of the
pulse. We derive an exact expression for the 6th order coefficient, C, and from
this deduce conditions under which this 6th order dependance is observed. We
also present new pulse sequences which achieve the same fidelity.
