Figure 4: Detection of bit-flip errors.

(a) Sequence used to assess performance of bit-flip QEC. After encoding by gates, either coherent (θ∈[0,π]) or incoherent (θ=0 or π) errors are introduced with single-qubit bit-flip probability p_err. Next, parallelized stabilizer measurements are either performed or replaced by an equivalent idling period. Partial tomography at this point is used to obtain the three-qubit fidelity F_3Q and the logical fidelity F_L. The calculation of F_L assumes incoherent second-round errors with the same p_err and a perfect decoding (dashed boxes). (b) Three-qubit fidelity F_3Q as a function of p_err with and without QEC under two scenarios: coherent errors applied on D_m (squares) and on all data qubits (circles). Standard deviations (0.005 for squares, 0.004 for circles) estimated by bootstrapping are smaller than the symbol size. The dashed line indicates the fidelity ceiling imposed by encoding errors. (c) F_L as function of p_err, obtained from the same data as in b. The average standard deviation of 0.005 is smaller than the symbol size. The individual contributions of the six cardinal states to F_3Q and F_L are shown in Supplementary Fig. 7. (d) F_L for all combinations of one and zero incoherent errors on all data qubits before and after QEC or idling. Error combinations are labelled m/n, with m (n) the number of errors before (after) QEC or idling. The case 1/1 is divided in two: errors on the same data qubit (1/1a) or on different qubits (1/1b).