Sixth-order robust gates for quantum control

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-Cε ⁶, where ε is the fractional error in the length of the pulse. We derive an exact expression for the sixth-order coefficient C and from this deduce conditions under which this sixth-order dependence is observed. We also present pulse sequences which achieve the same fidelity.