How quantum error correction detects and corrects errors without measuring the logical qubit
3-Qubit Bit-Flip Code
|0_L⟩ = |000⟩, |1_L⟩ = |111⟩
g₁ =Z₁Z₂ ⊗ I₃+1
g₂ =I₁ ⊗ Z₂Z₃+1
Syndrome Table
Error
g₁
g₂
→ Fix
none
+1
+1
✓ OK
X₁
−1
+1
→ apply X₁
X₂
−1
−1
→ apply X₂
X₃
+1
−1
→ apply X₃
Each syndrome measurement gives ±1 without revealing whether the logical qubit is |0_L⟩ or |1_L⟩.
Physics
A stabilizer is an operator g such that g|ψ⟩ = +|ψ⟩ for the code subspace. Any error E either:
• commutes with g (undetectable, but rare)
• anticommutes → eigenvalue flips to −1 (detected!)
The ±1 eigenvalue is the syndrome. It points to the qubit location without revealing the encoded information. This is the key miracle of quantum error correction.