Fig. 3: Z-rotation gate using Stark shifts.

a Order of magnitude of Stark shift δg_q, with respect to the bulk value g_Si ≈ 2.0, as a function of external magnetic field B₀ and single-qubit gate time τ_1Q, required for the π-rotation gate Z(π). b Gate layout realisation of the g-factor tuning scheme with shuttling through quantum dots under gates q − 1, to q, and to q + 1. The voltage V_q is used to tune the g-factor at site q, while the voltage V_μ is used to compensate for the change in the electrochemical potential due to the g-factor tuning. c, d Overlaid stability diagrams of the (q − 1, q, q + 1) triple quantum dot at the start and end (blue lines), and at the middle (red lines) of the shuttling sequence, illustrate the requirement for electrochemical potential compensation. Using a waveform as shown in Fig. 2e, shuttling proceeds from the charge configuration (n_q+1 n_q n_q−1) = (001) (blue circle marker) to (010) (red star marker), and to (100) (blue triangle marker). c In the perfectly compensated case with g-factor tuning, the (010) region opens up during the shuttling sequence. d In the non-compensated case, adjusting V_q to tune the g-factor at q causes the (010) region to shift away from the ground state charge configurations. e Electric field gradients evaluated along the cut shown as a grey dotted line in (d). Electric field gradients due to V_q are denoted as blue, and those due to V_μ as orange traces. f Estimated fidelity of Z(π) as a function of variance in actual gate duration and voltage noise affecting δg, with fixed σ_G = 10⁻³g_Si, B₀ = 1 T, and τ_1Q = 1 μs. (see main text).