Stabilizer Boundary Depth Scans

Lookup Distance Boundary Three-Row Scan

Correction lookup accepts no flip and one flip, then rejects two flips. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Lookup acceptance is a one-flip budget

The correction table can repair up to 1 flipped carrier.

m=1−dm=1-d
No-flip lookupThe zero syndrome leaves one unit of budget.correction lookup

Three lookup rows bracket the correction budget

The single-flip row is the boundary. The double-flip row is beyond the table contract.

ddmax⁡mgate011pass110pass21−1fail\begin{array}{c|c|c|c}d&d_{\max}&m&\text{gate}\\0&1&1&\text{pass}\\1&1&0&\text{pass}\\2&1&-1&\text{fail}\\\end{array}
Boundary lookupOne flipped carrier is still repairable.correction lookup

The double-flip row fails the one-flip contract

The helper computes the same repaired word that makes the accepted bit low.

m=−1failm=-1\quad \text{fail}
Double-flip lookupTwo flipped carriers exceed the contract.correction lookup