Fig. 4: Implementation and characterization of a Hopping Gate.

a Hopping sequence and sketched evolution of the spin state vector. The dashed (solid) arrow indicates the initial (final) vector at each step while yellow (blue) correspond to the original (secondary) quantum dot. The sequence results in a 90 deg rotation around the x-axis in D₁. b Spin-charge energy level diagram for the D_1,2 double-dot system illustrating the charge avoided-crossing. c Components of the effective spin qubit Hamiltonian \(H{\prime}\) with respect to the quantization axis of D₁ in the subspace of the charge ground state given by \(\langle {\sigma }_{i}\rangle={{{\rm{Tr}}}}({\sigma }_{i}H{\prime} )\) for the D_1,2 double-dot system (see Supplementary Note 6). d Experimental (left) and fitted (right) odd-parity probability using two cycles and four repetitions (refer to (a)) while sweeping t₁ and t₂ for the D_1,2 system with t_add fixed to 3.6 ns. The quantization axis tip is fit to 37.3(2) deg (see Methods for details). The red regions highlight the t₁ and t₂ where each one of the four repetitions achieves a high fidelity X₉₀ gate. The tunnel coupling for this measurement was estimated to be roughly 18 μeV. Additional data, fits and fidelity contours for qubits Q₁ and Q₄ are shown in Supplementary Fig. 4. e Fine calibration of t₁ by repeating the roughly calibrated gate many times with minor adjustments to one of the timing parameters. We extract the periodicity of the pattern and select the t₁ that matches a periodicity of four for the benchmarked gate. The difference in t₁ with respect to the data in (c) is due to adjustments in t_add. f Randomized benchmarking using both initial basis states. The Clifford set is compiled using only X₉₀ hopping gates and physical Z-rotations (see Methods). The fitted Clifford gate fidelities are \({F}_{{{{\rm{Clif}}}}}^{{{{\rm{hop,odd}}}}}=99.01(11) \% \) and \({F}_{{{{\rm{Clif}}}}}^{{{{\rm{hop,even}}}}}=99.49(7) \% \) for the odd and even parity inputs respectively (see Methods). g PIRS-like measurements showing the phase accumulation after a hopping gate and a 10 GHz burst when included in an echo sequence as depicted in Fig. 2b. The hopping gate is implemented as an X₁₈₀ gate on Q₄. The microwave burst carries the same energy as an average X₉₀ gate in this device during EDSR operation. Raw data is shown in Supplementary Fig. 10.