Lookup Distance Boundary Three-Row Scan - Quantum Mechanics V: Error Correction and Stabilizer Checks Step by Step

The lookup table has a correction-distance contract: it can repair only the number of flipped carriers it was built for. In this scan, the one-flip row sits on the accepted boundary, while the double-flip row is beyond the one-flip budget and fails even though the helper can still display a repaired word. Real QEC lookup tables also face measurement errors, correlated faults, decoder assumptions, finite rounds, and uncertainty.

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-d

Three lookup rows bracket the correction budget

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

\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}

The double-flip row fails the one-flip contract

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

m=-1\quad \text{fail}