Three-pulse amplitude-error compensation ($1-\mathcal{F}=3{\pi }^4{ϵ}^4/128+O({ϵ}^6)$)

For chronological equatorial rotations $R_{\varphi }(\pi )R_{2\varphi }(\pi )R_{\varphi }(\pi )$, the ideal propagator is $iX$. If all pulse angles have the same fractional error $ϵ$, choose $\cos \varphi =-1/2$ to cancel the quadratic term of the trace-overlap gate fidelity. With $x=\pi ϵ/2$, its exact fidelity is $|\cos x(1+{\sin }^{2}x/2)|$, whose leading infidelity is $3x^4/8$. This correction compensates a systematic amplitude error, not arbitrary axis or timing errors.