Perfect syndrome packing and decoder inputs

One theorem

For the exported generator order, the fifteen nonidentity single-qubit Pauli faults have all fifteen nonzero four-bit syndromes exactly once. Perfect readout therefore admits an exact lookup recovery, while circuit faults require additional flag or time-history information.

Complete syndrome table

The syndrome tuple lists anticommutation with (XZZX_, _XZZX, X_XZZ, ZX_XZ):

reproduce.py derives this table from the constructor and compares the set of values with all nonzero four-bit tuples. The table is convention-sensitive: reordering stabilizers permutes syndrome columns but preserves bijectivity.

Why “perfect” is exact packing

The codespace has dimension two. Acting with identity or one of fifteen single-site Paulis creates sixteen mutually orthogonal syndrome sectors of equal dimension. Their total dimension is

2 * 16 = 32,

which equals the entire five-qubit Hilbert space. Equivalently, the nondegenerate radius-one quantum Hamming count is

2^1 [C(5,0) + 3 C(5,1)] = 2^5.

There is no unused syndrome sector. Perfection is therefore a statement about this error set and nondegenerate packing, not about arbitrary circuit noise.

Exact perfect-readout recovery

The ideal decoder consumes four trustworthy syndrome bits:

1. 0000 maps to identity.
2. Any nonzero syndrome selects the unique table row.
3. Apply the same Pauli again; since a Pauli squares to identity up to phase, the state returns to the codespace with its logical information intact.

No likelihood model or tie-break is needed. Laflamme and collaborators describe the sixteen alternatives at decoder output (primary source); the Atlas recomputes their bijection in the exported stabilizer convention.

Why extraction faults are different

The ideal table assumes the syndrome is a reliable classical input. During a real extraction circuit, an ancilla or two-qubit-gate fault can both corrupt a reported bit and spread to multiple data qubits. The static four-bit address does not identify how that correlated fault arose.

A flag circuit adds one or more outcomes designed to fire on propagation paths that could create a dangerous data error. Its recovery table consumes the flag pattern, ordinary syndrome, and sometimes a follow-up extraction. Chao and Reichardt give a two-ancilla protocol for the five-qubit block (primary source); Bhatnagar and collaborators optimize flag-style depth (primary source).

A repeated circular extraction creates a time-ordered detection-event graph. Antipov and collaborators apply matching to that circuit-specific history (primary source). Its edge weights and event definitions do not transfer to an arbitrary schedule.

Decoder decision table

As of 2026-08-18, no model-independent asymptotic threshold or scalable decoder is assigned to the isolated block. Exact effective-channel or concatenation analyses remain valuable, but their channel, recovery, and level count are part of the claim.