Shor's 9-Qubit Code | mysimulator.uk

Error Channels

Bit-flip rate p_X (per qubit) 10%

Phase-flip rate p_Z (per qubit) 10%

Encoded logical state

This Round

Press "Inject Errors + Correct" to run a round.

Cumulative Statistics

Rounds run

Physical error rate

—

Logical bit-flip rate

0.0%

Logical phase-flip rate

0.0%

How it works

Shor's code (1995), the first quantum error-correcting code, protects one logical qubit against any single-qubit error by concatenating two 3-qubit repetition codes:

|0⟩_L = (|000⟩+|111⟩)(|000⟩+|111⟩)(|000⟩+|111⟩) / 2√2
|1⟩_L = (|000⟩−|111⟩)(|000⟩−|111⟩)(|000⟩−|111⟩) / 2√2

Inner code — bit-flip (X) protection. Each of the 3 blocks is a 3-qubit repetition code. Measuring Z₁Z₂ and Z₂Z₃ inside a block reveals which single qubit (if any) suffered an X error, without collapsing the encoded data — majority vote flips it back. A block only mis-corrects if 2 or 3 of its qubits are bit-flipped simultaneously.

Outer code — phase-flip (Z) protection. A single Z error anywhere inside a block flips that whole block's relative sign between |000⟩ and |111⟩ (an even number of Z errors in the same block cancels out). Comparing signs across the 3 blocks with X_1..6 / X_4..9 stabilizers is exactly a 3-qubit repetition code again, this time in the X basis — majority vote across blocks corrects the flipped one.

Any single-qubit error (X, Z, or Y = iXZ) touches
only ONE of the 9 physical qubits, so it always shows
up in exactly one channel and is always correctable.

p_X, p_Z sliders — independent per-qubit probability of a bit-flip / phase-flip error on each of the 9 physical qubits this round.
Inject Errors + Correct — samples one round of errors, runs both decoders, and shows whether each channel was corrected.
Run 1000 Trials — repeats silently to build a logical error rate, which stays far below the physical rate as long as p is small — the whole point of error correction.

Clean qubit Bit-flip (X) error Phase-flip (Z) error

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