Quantum Error Correction: 3-Qubit Bit-Flip Code

Encode logical qubit, detect & correct single-qubit errors via syndrome measurement
|0⟩_L = |000⟩  |  |1⟩_L = |111⟩  |  Syndrome: s₁=Z₁Z₂, s₂=Z₂Z₃  |  P(fail) = 3p²−2p³

Logical Qubit & Physical Encoding

q₁
q₂
q₃
(click to inject error)
Syndrome bits:
s₁ = Z₁Z₂ = ?   s₂ = Z₂Z₃ = ?
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Error Rate vs. Logical Failure

Logical error rate: —   Uncoded: —

Event Log

3-qubit bit-flip code: Encodes one logical qubit into three physical qubits. Logical |0⟩_L = |000⟩, |1⟩_L = |111⟩. A single bit-flip error (X gate) on qubit i flips two syndrome bits — the syndrome pattern uniquely identifies the error location.

Syndrome measurement: Measure Z₁Z₂ and Z₂Z₃ (stabilizer generators). Results (0,0)=no error; (1,0)=qubit 1; (1,1)=qubit 2; (0,1)=qubit 3. Crucially, syndrome measurement does NOT collapse the logical qubit state — it projects onto the error subspace.

Threshold theorem: If physical error rate p < p_threshold, logical error rate P_L = 3p² − 2p³ < p. For the surface code, p_threshold ≈ 1%. Shor (1995) showed fault-tolerant QC is possible.