Fig. 4: Two-excitation QST in 2D qubit systems with optimized couplings.

a Measured couplings of the 3 × 3 qubit network for the optimized two-excitation QST (see Supplementary Fig. 7 for the specific values). b The corresponding experimental time evolution of two-excitation state QST in Fock space \([{{{{{{\mathcal{D}}}}}}}_{{\hat{H}}}=({9\atop2})=36]\), where each marker denotes a Fock state—here, we contrast a solution for QST-optimized couplings from one with randomly chosen J_m,n at different representative times. The concentric circles denote the Fock states with the same distance from the initial state. c The dynamics of the average distance 〈d(t)〉 traveled in Fock space for both cases; the dashed (dotted) line gives the maximum (mean) distance. d Measured couplings of the 6 × 6 qubit network for the two-excitation QST \([{{{{{{\mathcal{D}}}}}}}_{{\hat{H}}}=t({36\atop2})=630]\) after optimization (see Supplementary Fig. 8 for the specific values). e The (Q₁, Q₂) and (Q₃₅, Q₃₆) populations over time using the measured couplings in (d), which yield a transfer fidelity of 0.737 ± 0.007 at about 250 ns. Error bars in (c) and (e) come from the standard deviation of five experimental repetitions. f Numerically computed distribution of the ratio of adjacent gaps P(r) in the case of QST-optimized and random couplings. Here, we take an average of an ensemble of k = 40 coupling matrices to improve statistics; dashed and dotted lines are surmises for the Wigner–Dyson and Poisson distributions⁴⁸, respectively (see Supplementary Note 7). t_ΔE is a minimum time for a perfect QST set by “quantum speed limit” arguments (see Methods).