3-Qubit Bit-Flip Code | mysimulator.uk

Encode Logical Qubit

Noise Channel

Bit-flip probability p 10%

Run

Syndrome & Decode

s₁=0 s₂=0 → no error detected

Physical qubits

0 0 0

Decoded logical

|0⟩_L

Statistics

Rounds run

Logical errors

Empirical P(fail)

—

Theory 3p²(1−p)+p³

0.3%

How it works

The 3-qubit bit-flip code encodes one logical qubit into three physical qubits by repetition:

|0⟩_L = |000⟩
|1⟩_L = |111⟩

Each round, an independent bit-flip (Pauli-X) channel acts on every physical qubit with probability p — the same noise model used to model relaxation errors on real superconducting/trapped-ion hardware. The code does not measure the qubits directly (that would destroy superposition); instead it measures two stabilizer parities that reveal only the error location:

s₁ = Z₁Z₂  (parity of qubits 1,2)
s₂ = Z₂Z₃  (parity of qubits 2,3)

A single flip on qubit i produces a unique syndrome, so a lookup table corrects it exactly:

s₁ s₂ → correction
0  0  → none
1  0  → flip qubit 1
0  1  → flip qubit 3
1  1  → flip qubit 2

This decoder fails only when two or three qubits flip in the same round — two flips get miscorrected to the wrong codeword, and three flips return a valid but wrong codeword the syndrome cannot see. The resulting logical error probability is:

P_L = 3p²(1−p) + p³

For any p < 1/2 this is smaller than the uncorrected single-qubit error rate p — the mechanism behind every real quantum error-correcting code (surface codes, color codes) used to make quantum computers fault-tolerant. Toggle Correction: OFF to see physical errors accumulate unchecked instead.

Physical qubit |0⟩ Physical qubit |1⟩ Syndrome ancilla (lit = mismatch)

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3-Qubit Bit-Flip Code — Quantum Error Correction