Quantum Error Correction: Stabilizer Codes

3-qubit bit-flip code — Syndrome measurement without state collapse

Stabilizer Generators

S₁ = Z₁Z₂ = +1

S₂ = Z₂Z₃ = +1

Syndrome: (s₁,s₂)

( +1, +1 )

Diagnosis:

No error

Logical State

|0̄⟩ = |000⟩

|q₁q₂q₃⟩ = |000⟩

Correction:

Identity (no error)

Inject Error

The 3-qubit bit-flip code encodes |0̄⟩ = |000⟩, |1̄⟩ = |111⟩. Stabilizer generators Z₁Z₂ and Z₂Z₃ detect which qubit flipped without revealing the logical state. Syndrome (−1,+1) → error on q₁, (+1,−1) → q₃, (−1,−1) → q₂.