Fig. 4: Coherence improvement.

Black, blue, and green circles are experimentally measured state fidelities at a given time. The fidelities are extracted from the tomographic reconstruction of states with 5000 repeated measurements. Error bars (one standard deviation) are smaller than the marker size³⁸. The improvement with AQEC turned on is explained by the master equation simulation. a\(\left\vert {L}_{0}\right\rangle\) and b\(\left\vert {L}_{1}\right\rangle\) traces are fitted to the exponential decay curve \(A\exp (-t/\tau )+C\), and c\(\left\vert {L}_{x}\right\rangle\) traces are fitted to \(A\exp (-t/\tau )\). The offset C is necessary since fidelity will achieve steady-state values. The error (one standard deviation) for τ are obtained from the fitting. The large uncertainty comes from treating C as a free variable in the fitting. The fast transition period (first \(1.5\,\mu {{{{{{{\rm{s}}}}}}}} \sim {\Omega }_{j}^{-1}\) in the AQEC case is not included in the fitting^* for a better representation of logical coherence. ^*First 10.5 μs data are used in \(\left\vert {L}_{1}\right\rangle\)’s free decay case for better fitting.