Single qubit gates with jump and return sequences

We discuss the implementation of frequency selective rotations using sequences of hard pulses and delays. These rotations are suitable for implementing single qubit gates in nuclear magnetic resonance (NMR) quantum computers, but can also be used in other related implementations of quantum computing. We also derive methods for implementing hard pulses in the presence of moderate off-resonance effects, and describe a simple procedure for implementing a hard 180° rotation in a two spin system. Finally, we show how these two approaches can be combined to produce more accurate frequency selective rotations.

Figure 1

Jump and return solvent suppression sequence: The solvent signal (at zero frequency in the rotating frame) experiences no overall evolution, while spins at other frequencies are partially excited.

Figure 2

Propagator fidelity for a selective 90° pulse implemented using the jump and return approach of Eq. (4) as a function of the ratio $g$ between the actual offset frequency and its nominal value. The solid line shows the fidelity with respect to a 90° pulse, while the dashed line shows the fidelity against an identity operation. These curves are given by $\cos [\pi (1\mp g)∕8]$, respectively.

Figure 3

Off-resonance effects on a 90° excitation pulse for off-resonance fractions $f$ in the range 0 to 1 in steps of 0.1.