Extended Data Figure 1: Deviations from sin²(•) behaviour.

Colouring as in Fig. 2. a, The fidelity as a function of duration for the optimal families rescaled by the apparent QSL. The T_QSL is found for each solution by fitting sin²(aT + b) where a and b are fitting parameters. The black line shows sin²(π/2T) for reference. The inset shows the infidelity (1 − F) as a function of duration close to the . b, The direct Hilbert velocity 〈Q〉_T for the best solutions found by the HILO (purple), tunnelling (yellow) and shovelling (blue) clans respectively. Note that 〈Q〉_T is only defined for durations with F < 1, so the curves end at different durations. The varying direct Hilbert velocity explains the deviation from in a.