Figure 4

Higher-order WAMFs. Construction of \(\mathrm{WAMF}_{0:7}^{(2)}\) for dephasing noise filtering. (a) Representative amplitude-modulated profiles for spectrally-optimized 8-segment \(\mathrm{WAMF}_{0:7}^{(2)}\) gates. Vertical axes indicates Rabi rate values \(\Omega_{l}\) in units of \(1/\tau\) for the 8-segments. (b) Corresponding (Paley ordered) Walsh spectra. Vertical axes indicate values of the Walsh spectral amplitudes \(X_{k}\) in units of \(1/\tau\). Optimized spectra obtained via Nelder-Mead search. (c) Log-scale color plot of the cost function \(A_{z}(X_{5},X_{6})\) (integrated over \(\omega\in[10^{-9}, 10^{-1}]\tau^{-1}\)) defined on representative two-dimensional cross section of \(\mathbf{X}_{\nu}\)-domain. Blue regions indicate minima in \(A_{z}(X_{5},X_{6})\), implying second-order optimized filter synthesis. ‘Cross-region’ (circled) indicates robustness region with respect to errors in \(X_{5,6}\). (d) Dephasing filter-transfer functions for the optimized \(\mathrm{WAMF}_{0:7}^{(2)}\) gates in (a), compared against primitive \(\pi_{x}\) rotation and optimized \(\pi_{x}\) \(\mathrm{WAMF}_{0,3}^{(1)}\) gate. For the blue, red and green traces the cost function \(A_{z}(X_{3},X_{5},X_{6})\) was defined over the band \([\omega_{L},\omega_{c}]\) with \(\omega_{c} = \tau^{-1}\) and \(\omega_{L} = (10^{-4},10^{-3},10^{-2})\tau^{-1}\).