Logical operations and realization boundaries

One theorem

For the exported logical basis, uniform physical H implements logical (H tensor H) SWAP, uniform physical S implements logical CZ up to Pauli phases, and pairwise physical CNOT between aligned blocks implements parallel logical CNOT on the corresponding logical pairs.

These are exact Pauli-conjugation statements. Whether a laboratory circuit is fault tolerant additionally depends on preparation, routing, extraction, and postselection.

Uniform physical H

Conjugation swaps X and Z on every physical qubit. Modulo the stabilizers,

For example, physical H sends XX__ to ZZ__; multiplying by the stabilizer ZZZZ gives __ZZ. Thus each logical qubit receives H and the two logical labels are exchanged. Meier, Eastin, and Knill give the same logical action in their chosen basis (primary source).

Uniform physical S

Ignoring the tracked Pauli phase, S X S^dagger = XZ and Z is fixed. Hence

Xbar_1 -> Xbar_1 Zbar_2, and

Xbar_2 -> Zbar_1 Xbar_2,

while both Z logicals stay fixed. This is exactly the Clifford conjugation table of logical CZ. The reproduction script performs the binary symplectic calculation modulo stabilizers rather than relying on these displayed representatives.

Inter-block physical CNOT

Under CNOT, control X acquires target X and target Z acquires control Z. Apply the gate pairwise between two equally indexed blocks. Every X stabilizer on the control acquires the corresponding target stabilizer, and every Z stabilizer on the target acquires the corresponding control stabilizer, so the joint stabilizer row space is preserved. The same rule acts on both logical pairs and gives two parallel encoded CNOTs.

The statement is transversal in an interaction graph that provides aligned pairs. It does not include motion, crosstalk, or correlated decoding.

Magic-state protocols

Meier, Eastin, and Knill use this code in an accepted-round routine that consumes ten noisy Hadamard eigenstates and outputs two improved states. The transversal Clifford symmetries explain why the block is useful, but the distillation claim also includes the preparation, measurements, acceptance condition, and input-noise assumptions.

Gupta and collaborators prepare an encoded CZ resource in the four-qubit block and then deform and teleport it into a one-logical-qubit code (primary source). The code transition is a protocol boundary: the reported beyond-break-even fidelity cannot be assigned to bare XXXX/ZZZZ storage.

What the experiments establish

• Linke and collaborators use four trapped-ion data qubits and an ancilla to demonstrate fault-tolerant preparation, measurement, and operation of one logical qubit. The other encoded degree of freedom is a gauge qubit, so the protocol does not claim arbitrary protected two-logical-qubit computation.
• Takita and collaborators demonstrate fault-tolerant preparation in a protected one-logical-qubit subspace on superconducting hardware. Again, the subsystem choice is part of the result.
• Vuillot and Roffe and collaborators show postselection benefits on IBM five-qubit devices using different circuit representations. Harper and Flammia benchmark logical Clifford gates and report an encoded-gate benefit, while also explaining that state preparation and measurement prevent an end-to-end below-threshold conclusion.

Primary records: Linke et al., Takita et al., Vuillot, Roffe et al., and Harper–Flammia.

What the experiments do not establish

They do not give this fixed block an asymptotic threshold. A threshold needs a growing family plus a specified noise model, syndrome circuit, decoder, and observable. Criger and Terhal study such a tuple for a distinct [[4,2,2]]-concatenated Toric construction (primary source); their threshold does not transfer to this entry.