Detection, correction, and erasure side information

One theorem

The exported [[4,2,2]] block detects the complete set of single-qubit Pauli faults, cannot correct that set when the position is unknown, and can correct an arbitrary error on one position when that position is supplied as erasure side information.

These are three statements about three different error sets.

The complete syndrome table

Order the syndrome bits as the anticommutation outcomes against (XXXX, ZZZZ). For every physical position i,

All twelve nonidentity weight-one errors have nonzero syndrome, so all are detected. The table has only three distinct nonzero rows: moving the error does not change the result.

Why a correction guess fails

For a correctable set {E_a}, the Knill–Laflamme condition requires

P E_a^dagger E_b P = c_ab P,

where P projects onto the codespace. Consider two same-type alternatives at different unknown positions, E_a=X_i and E_b=X_j. Their product is X_i X_j. It commutes with the stabilizers and, for suitable i,j, is one of the exported logical operators or differs from one by a stabilizer. Its restriction to the codespace is not a scalar. Therefore the complete unknown-position single-fault set violates the condition.

The syndrome collision and the theorem say the same thing in two languages. If a lookup table guesses i but the error was at j, recovery leaves X_i X_j. A nonzero detector outcome tells us to reject; it does not license an arbitrary correction.

This is why the trapped-ion experiment uses postselection and explicitly states that the code cannot locate an unknown error (Linke et al.).

Why known erasure is different

Now suppose a classical record says position i was erased. The error set is only {I,X_i,Y_i,Z_i}. Every product E_a^dagger E_b is either identity or a Pauli supported at that same position. The nonidentity products have nonzero syndrome, so their projection into the codespace vanishes; identity projects to P. The Knill–Laflamme condition holds.

The recovery map must consume both pieces of information:

1. the known physical position, supplied by the channel or a conversion mechanism; and
2. the parity information needed to reconstruct the state.

Grassl, Beth, and Pellizzari define the erasure channel precisely as an arbitrary error at a known position and give the four-qubit encoding (primary source). Gingrich and collaborators use the two-to-four encoding in an optical loss-memory proposal (primary source). Neither source turns the unknown-position Pauli detector into a one-error correcting code.

Measurement faults are a fourth problem

The table above assumes perfect measurement of both global checks. A faulty ancilla or entangling gate can corrupt the syndrome and propagate errors into the data. A circuit-level detector therefore requires a preparation circuit, gate ordering, ancilla verification or repeated checks, and a rule for the resulting time history.

Gottesman frames small fault-tolerance experiments around precisely this protocol-level criterion (primary source). The static block is an ingredient. “Distance two” alone neither certifies nor refutes a particular fault-tolerant preparation or postselected computation.

Practical decision table

This table is the boundary carried into card.yaml: the two lookup rows name different inputs and do not pretend to be one universal decoder.