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Quantum Error Correction

Quantum

The 3-qubit bit-flip code: one logical qubit is spread across three physical qubits, a bit-flip error is injected on any line, and two ancilla qubits extract a syndrome that names the culprit — without ever measuring the data. A majority-vote correction restores the logical state and the fidelity returns to one.

6 components · 13 connections · stabilizer parity + syndrome decoder

the aha

You can protect a qubit you are not even allowed to look at — by measuring only whether the copies disagree, never what they say.

why it works
Measuring a qubit destroys its superposition, so the classical trick of "read it and fix it" is off the table. The bit-flip code sidesteps this beautifully. Encode α∣0⟩+β∣1⟩\alpha|0\rangle + \beta|1\rangle as α∣000⟩+β∣111⟩\alpha|000\rangle + \beta|111\rangle — the amplitudes are untouched, just spread across three qubits. Now measure two parities: s0=q0⊕q1s_0 = q_0 \oplus q_1 and s1=q1⊕q2s_1 = q_1 \oplus q_2. A parity reveals whether two qubits differ, never their actual values, so the delicate α,β\alpha, \beta survive. Yet the syndrome (s0,s1)(s_0, s_1) points straight at the flipped qubit: 1010 means q0q_0, 1111 means q1q_1, 0101 means q2q_2, 0000 means clean. Flip it back and the logical state is exactly restored. This is the whole reason scalable quantum computing is believed possible.
governing model
∣0⟩L=∣000⟩, ∣1⟩L=∣111⟩  ⋅  s=(q0⊕q1, q1⊕q2)|0\rangle_L = |000\rangle,\ |1\rangle_L = |111\rangle \;\cdot\; s = (q_0{\oplus}q_1,\ q_1{\oplus}q_2)
structural inventory
  • logical qubit →3\to 3 physical
  • injectable XX error
  • 22 ancilla syndrome bits
  • majority-vote decoder
boundaries
  • corrects any single bit-flip
  • syndrome ∈{00,01,10,11}\in\{00,01,10,11\}
  • logical fidelity 0/10/1
fault relay syndrome decoder miswired

  1. Inject a bit-flip on any one line.

    Pick q0q_0, q1q_1, or q2q_2. The physical state is now corrupted — but you are forbidden from reading the data to find out how.

  2. Read the syndrome, not the data.

    The two parity bits light up a unique pattern that names the guilty qubit while learning nothing about α,β\alpha, \beta. Watch the logical fidelity snap back to 11 after correction.

  3. Miswire the decoder.

    Corrupt the syndrome→correction map and the code "fixes" the wrong qubit, driving fidelity to 00. A correct decoder is as essential as the code itself.
causal signal trace paused

Prepare One logical qubit enters the encoder.

step 1 / 6 Prepare |ψ⟩

Loading Quantum Error Correction model…