Extended Data Fig. 2: Illustration of error recovery and frame repair procedures.

We illustrate the procedure for the surface code, where a cross-sectional view with one spatial axis and one time axis is shown. We only illustrate X errors and Z stabilizer measurement errors, which are relevant to interpreting the \(\overline{Z}\) measurement. X errors can terminate on orange boundaries, but cannot terminate on cyan boundaries. The transversal \(\overline{CNOT}\) copies X errors from the top to the bottom, resulting in a branching point (black cross) and an error cluster spanning both code blocks. (a) Error chains and frame flips. Chains of X-type errors (orange lines) lead to syndromes (end points) or terminate on appropriate boundaries. A line segment in the vertical direction is a data qubit X error, while a line segment in the horizontal direction is a measurement error. Note that the X-type error cannot terminate on the transversal Z measurement boundary. The random stabilizer initialization leads to a frame configuration on the logical \(| \,\overline{+}\,\rangle \) initialization, as illustrated by the blue line and the flipped Z stabilizer (blue point). This is similar to the frame stabilizer operator g_s illustrated in Extended Data Fig. 1(a). (b) We first infer an error recovery operator, which has the same boundary as the error chain. Together, the error and recovery operator form the fault configuration, which triggers no detectors. We illustrate a few examples (orange lines) that do not lead to a logical error: (1) the fault configuration forms a closed loop and is equivalent to applying a stabilizer; (2) the fault configuration terminates on an initialization boundary; (3) the fault configuration terminates on an out-going, unmeasured logical qubit, but the forward-propagated errors onto the measured logical qubit are equivalent to a stabilizer. A logical error can only happen when the fault configuration spans across two opposing spatial boundaries (red line), which requires an error of weight Θ(d). (c,d) The frame repair operation returns the logical qubit to the code space with all stabilizers +1, corresponding to cancelling any residual flipped stabilizers on the initialization boundary. Note that the error recovery process may also lead to a change that needs to be accounted for by frame repair. An example choice of frame repair is shown in (c), which applies an overall X operator on the logical measurement result. Alternatively, a different choice of frame repair shown in (d), related to the previous one by a frame logical flip, results in identity operation on the logical measurement result.