Syndrome Table Capacity Boundary Scan - Quantum Mechanics V: Error Correction and Stabilizer Checks Step by Step

Syndrome table capacity is fixed by how many checked parity bits are available. With two syndrome bits, the table has four rows, so adding the zero-syndrome row fills the table and treating a double-error class as new demand overfills it. Real codes must account for aliasing, correlated errors, decoder choice, repeated rounds, calibration drift, and uncertainty rather than assuming every error gets a unique row.

Two syndrome bits make four table rows, so double-error aliases cannot be new rows. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

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Two syndrome bits give four table rows

The table capacity is fixed by the two checked parity bits.

N_{\text{rows}}=4

Demand rows bracket the syndrome table capacity

Adding the zero-syndrome row fills the table. Treating a double error as new demand overfills it.

\begin{array}{c|c|c|c}N_{\text{demand}}&N_{\text{rows}}&m&\text{gate}\\3&4&1&\text{pass}\\4&4&0&\text{pass}\\5&4&-1&\text{fail}\\\end{array}

A double-error class aliases an existing row

The lookup helper shows the double-error syndrome is not a new independent table row.

m=-1\quad \text{fail}