Question. If the perfect-syndrome decoder is only a seven-row table, where does the real fault-tolerance difficulty enter?
The table answers “which data correction matches these bits?” An extraction gadget answers the prior question “why should these bits be trusted after one fault?” Treating the two as one algorithm hides the dominant engineering problem.
For a Z error on qubit j, X-check row i anticommutes exactly when H[i,j]=1. The three-bit X-check syndrome is therefore column j:
| zero-indexed qubit | syndrome | X-checks that fire |
|---|---|---|
| 0 | 001 | row 2 |
| 1 | 010 | row 1 |
| 2 | 011 | rows 1, 2 |
| 3 | 100 | row 0 |
| 4 | 101 | rows 0, 2 |
| 5 | 110 | rows 0, 1 |
| 6 | 111 | rows 0, 1, 2 |
Apply Z to the addressed qubit. A single X error is located by the identical table read from the Z-check outcomes and is corrected with X. A Y error has both components, so both syndrome triples identify the same qubit. The reproduction script asserts that the seven columns are distinct and exhaust all nonzero three-bit strings.
Under a memoryless Pauli model and perfect measurement, maximum-likelihood decoding for a single fault therefore reduces to this lookup. For multiple faults, a three-bit syndrome can no longer identify a unique error: errors differing by a stabilizer have the same action, while errors in distinct logical cosets can also share the measured syndrome. A decoder must then use its noise model to choose a coset, and distance three offers no promise beyond one unknown-qubit error.
Measuring a weight-four check requires an ancilla to interact with four data qubits. CNOT propagation depends on direction:
X_control -> X_control X_target,
Z_target -> Z_control Z_target.An ancilla error of the propagating type, occurring before its remaining interactions, can fan out to two or more data qubits. The final ancilla measurement may still return a syndrome compatible with an ordinary one-qubit error. Because a distance-three block does not generically correct weight-two data faults, “measure each generator once and consult the table” is not a fault-tolerant recovery protocol.
The check matrix does not specify CNOT order, ancilla state preparation, verification, repetition, or classical stopping. Those choices must be named whenever a circuit-level result is quoted.
Verified cat states. Shor-style extraction distributes the check coupling over a cat ancilla so that one ancilla fault cannot silently fan out through a single reused qubit. Verification rejects dangerous cat-state faults before the data interaction. The price is extra ancillas, verification circuitry, and repeated measurement (Shor).
Flags. A small flag circuit is arranged so every single fault that could create a dangerous correlated data error also changes a flag outcome. The decoder key is then not merely the three-bit syndrome but the pair (syndrome, flag history). Chao and Reichardt give a two-extra-qubit Steane scheme (paper); Reichardt later develops low-ancilla parallel variants and exposes the latency/ancilla trade (paper).
Encoded ancillas. Steane-style error correction couples the data block to an encoded ancilla so X and Z syndrome information can be extracted in a blockwise fashion. Ancilla preparation and verification are themselves fault-tolerant subroutines. A trapped-ion experiment compared repeated encoded-ancilla recovery with flag extraction (Postler et al.); the comparison belongs to those concrete circuits and device noise, not to the abstract lookup table.
Data-syndrome coding and coherent measurement-free correction are further alternatives. They change which syndrome faults are encoded or whether mid-circuit measurement is used, so their numbers are comparable only after the full hardware and noise contract is aligned.
With noisy measurement, a single syndrome triple is insufficient. Repeating until two consecutive outcomes agree is a policy, not a theorem of the code; it trades latency against the chance of accepting a measurement fault. Adaptive flag protocols use the observed history to decide which checks to remeasure. Real-time experiments additionally include classical processing and feedback latency.
Consequently, a complete Steane decoder record has four layers:
The card's first lookup row describes the perfect-syndrome table. Its second row describes flag-aware circuit decoding. Keeping them separate prevents a perfect-measurement result from masquerading as circuit-level fault tolerance.
The fixed block has no parameter that increases distance. An asymptotic threshold statement requires a scaling sequence, such as concatenation level or growing color-code distance, plus a decoder and noise normalization. A one-level pseudo-threshold, logical error per recovery cycle, and concatenation threshold answer different questions.
The T11 decoder sweep, searched through 2026-07-26, found no source supporting an intrinsic asymptotic threshold for the bare [[7,1,3]] lookup decoder. That dated negative result is why card.yaml describes regimes and circuit assumptions but does not attach a threshold number to Steane itself.