Figure 3

First-order WAMFs. Construction of \(\mathrm{WAMF}_{0,3}^{(1)}\) for dephasing noise filtering. (a) Log-scale color plot of the cost function \(A_{z}(X_{3};X_{0})\) integrated over \(\omega\in[10^{-9}, 10^{-1}]\tau^{-1}\). Total gate angle \(\Theta= X_{0}\) (\(\tau\equiv1\)). Blue regions indicate minima in \(A_{z}(X_{3};X_{0})\), implying optimized filter synthesis. Coloured lines (blue, red, green) at \(X_{0} = (2\frac{1}{4}, 2\frac{1}{2}, 3)\pi\) correspond to rotation angles \(\theta= ( \frac{\pi}{4}, \frac{\pi}{2},\pi)\). These lines terminate at values \(X_{3} = (0.36\ldots, 0.65\ldots, 1)\pi \) on a blue contour (boxed) and indicate representative points in the \(X_{0}X_{3}\) plane for which first-order filtering is achieved. In this plot, for \(|X_{3}|>|X_{0}|\) the Rabi rates \(X_{\pm}\) have opposite sign, implying a π-phase shift in addition to amplitude modulation (e.g. see Eq. (145)). We therefore distinguish quadrants Q1 and Q3 in the \(X_{0}X_{3}\) plane in which \(|X_{3}|\le|X_{0}|\) (strict amplitude modulation) and Q2 and Q4 in which \(|X_{3}|>|X_{0}|\) (sign-switching amplitude modulation). (b) Solid lines: first order Taylor coefficient \(C_{2}^{(z)}(X_{3};X_{0})\) as a function of \(X_{3}\) with \(X_{0} = (2\frac{1}{4}, 2\frac{1}{2}, 3)\pi\); zeros appear as dips on log-scale. Dotted lines: one-dimensional slices of \(A_{z}(X_{3};X_{0})\) for same fixed values of \(X_{0}\). Boxed dips correspond to boxed points in a) where the colored lines intersect with the blue contour. (c) Filter-transfer functions for the spectrally optimized \(\mathrm{WAMF}_{0,3}^{(1)}\) gates identified by the boxed features in (a) and (b).