Past exam of the laser science and quantum information processing course of the University of Oxford / 2018 / 4 / d / Solution

An ideal $\pi$ pulse is $-i{\sigma }_{\varphi }$. Since ${\sigma }_{\varphi }{\sigma }_{2\varphi }{\sigma }_{\varphi }={\sigma }_x$, the chronological three-pulse sequence has propagatorIt implements the Pauli X gate up to global phase, for every $\varphi$.

For this three-pulse amplitude-error compensation, the leading error in the supplied fidelity expansion vanishes when $1+2\cos \varphi =0$, givingThese choices minimize the leading small-angle-error infidelity. Substituting $\cos \varphi =\cos 2\varphi =-1/2$ into the fourth-order term givesThe fourth-order coefficient is nonzero, so this three-pulse sequence improves the single pulse's quadratic infidelity to quartic infidelity, rather than sixth-order infidelity.

An exact check is available. With $x=\pi ϵ/2$, multiplication of the three imperfect rotation gates at either optimum givesIts Taylor expansion has precisely the fourth-order coefficient above. The improvement concerns the same systematic fractional angle error on all three pulses; other errors need not cancel.