Fig. 4

Randomized benchmarking of operations on encoded qubit. a Randomized benchmarking (RB) sequence. In RB a sequence of Clifford operations of length n is chosen at random (U_{X,Y,…}), followed by the operation which inverts the effect of the sequence (U_corr). In order to apply this technique to the operations on the encoded qubit, we begin the experiment by encoding, and decode before measurement. Our implementation of RB creates a new random gate sequence for every measurement, and is thus not biased by the distribution of sequences which are measured. b Interleaved randomized benchmarking (iRB) sequence: In order to establish the fidelity of a single operation (here, U_X), the operation is interleaved with random operations, and the benchmarking result is compared with the non-interleaved case. c The probability of measuring the correct result vs. sequence length n is fit to a two parameter model p _correct = 0.5 + Ae ^−n/τ. The lower panel shows the fit residuals. Each data point is the result of 2000 averages, with a new sequence realization every shot. The error averaged over all gates is computed as r = c(1−e ^−1/τ(RB))/2²⁶. The average error for a single gate X is computed as r(X) = c(1−e^{1/τ(X)−1/τ(RB)})/2²⁷. The factor c = 1.7 ± 0.1 is a correction factor to compensate for the underestimation of the error rate in the presence of leakage to a larger Hilbert space (Supplementary Note 4)