Reducing the sensitivity of the M{\o}lmer-S{\o}rensen gate for ion-trap quantum computing to unbalanced laser intensities

Geometric gates for ion-trap quantum computing have been experimentally demonstrated (fidelity $\approx 0.97$), using a single pair of laser beams driving a Raman transition between motional states\footnote{D. Leibfried et al., Nature 422, 412 (2003).}. This class of gate operations only works for qubits encoded in states with field-sensitive transitions. Since qubits encoded in states with field-insensitive transitions are less susceptible to decoherence, there is presently a renewed interest in the M{\o}lmer-S{\o}rensen (MS) gate, a geometric gate compatible with field-insensitive states\footnote{P. J. Lee et al., quant-ph/0505203}. A fundamental weakness of the MS gate is that it requires two Raman transitions to be driven simultaneously, introducing a new error source in the form of potentially unbalanced strengths of the two pairs of Raman beams. We show that although the MS gate in the originally proposed form is highly sensitive to such an imbalance, a minimal modification allows it to operate at high fidelity, even with poorly balanced laser intensities.