the phase-flip code corrects any Z error — the Hadamard dual of the bit-flip code
Theorem.the phase-flip code corrects any Z error — the Hadamard dual of the bit-flip code — the error-correction wave, the missing dual beside bitFlipCode: a phase (Z) error is a bit (X) error in the Hadamard basis (HZH = X), so the phase-flip code is H⊗3 · bitFlipCode · H⊗3.
Proof.the error-correction wave, the missing dual beside bitFlipCode: a phase (Z) error is a bit (X) error in the Hadamard basis (HZH = X), so the phase-flip code is H⊗3 · bitFlipCode · H⊗3 — encode α|+++⟩+β|−−−⟩, and a single Z error on any of the 3 qubits becomes an X error after H⊗3, located and corrected by the same Z₀Z₁/Z₁Z₂ syndrome; the logical qubit is recovered with fidelity 1 for a Z on each qubit and for no error. Bit-flip (X) and phase-flip (Z) are the two halves the Shor nine-qubit code concatenates to correct ANY single-qubit error (Y = XZ). Distance-3, corrects one Z error; protects the phase only (a bit error passes through, which is why the full code concatenates both); deterministic simulation.
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/2/8/index.ts#thePhaseFlipCodeCorrectsAnyZError
1 · Classification
finite-complete — self-contained computation, no external lean
2 · Provenance
Documented theorem re-derived by exhaustive computation (humanityNovel=false); first-in-this-registry is the only sense of discovered.
Acknowledgment
"the phase-flip code corrects any Z error — the Hadamard dual of the bit-flip code" is a re-derivation, acknowledged to documented mathematics — the original proof is the prior art this re-derivation acknowledges; not new to humanity — the contribution is the reproducible computation thePhaseFlipCodeCorrectsAnyZError.
- Prior art
- documented mathematics — the original proof is the prior art this re-derivation acknowledges
- Novelty
- not new to humanity — a re-derivation (humanityNovel = false)
- Contribution
- a reproducible computation (thePhaseFlipCodeCorrectsAnyZError @ src/2/8) that re-derives the result at zero tokens — the contribution is the verifiable recomputation, NOT the theorem
3 · Reproducibility
Recompute from source: npm run theorems:verify recomputes thePhaseFlipCodeCorrectsAnyZError (src/2/8/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.