Stabilizer Boundary Depth Scans

Round Repeat Boundary Three-Row Scan

Repeated syndrome rounds separate a one-round blip from a data error that deserves correction. This ledger ignores the baseline round, counts later nonbaseline readings, and accepts correction only when the nonbaseline syndrome repeats enough times to reach the threshold. Real stabilizer cycles also include readout mistakes, drifting ancillas, timing limits, finite sampling, and uncertainty.

Syndrome rounds need a repeated nonbaseline reading before data correction. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Data correction needs repeated nonbaseline rounds

The repeat count ignores the baseline round and counts later nonbaseline readings.

m=r−rmin⁡m=r-r_{\min}
Baseline roundsNo later nonbaseline reading is present.syndrome rounds

Three round patterns bracket the data-error threshold

A one-round blip is still below threshold. Two repeated nonbaseline readings meet it.

rrmin⁡mdecision02−2no error12−1check fault220data error\begin{array}{c|c|c|c}r&r_{\min}&m&\text{decision}\\0&2&-2&\text{no error}\\1&2&-1&\text{check fault}\\2&2&0&\text{data error}\\\end{array}
Check-fault boundaryOne nonbaseline round is not enough.syndrome rounds

The repeated row reaches the data-error threshold

The helper returns the correction index only when the nonbaseline syndrome repeats.

m=0passm=0\quad \text{pass}
Repeated syndromeTwo matching rounds support a data correction.syndrome rounds