Syndrome explorer

A stabilizer code never looks at the state it protects — measuring it would destroy the superposition. It measures only whether each of its generators was disturbed. Apply an error below and follow what happens: which generators fire, what the decoder concludes from that alone, and whether the correction restored the logical state or silently changed it.

Code

Steane [[7,1,3]]CSS, and the syndrome of a single X error is the binary index of the qubit it hit.Five-qubit perfect [[5,1,3]]The smallest code correcting any single-qubit error, and not CSS.Shor [[9,1,3]]Concatenated: six generators watch for bit flips, only two for phase flips.Three-qubit repetition [[3,1,1]]Corrects any single bit flip, yet has distance 1: no generator can see a Z error.

Error

One letter per qubit: I, X, Y or Z.

XIIa single bit flip ZIIa single phase flip YIIboth at once XXItwo errors at once ZIIcommutes with every generator, so nothing is measured ZZIacts as the identity on the code space

Three-qubit repetition code, error ZII

Undetected — silent logical error

This error commutes with every generator, so the syndrome is blank and the decoder does nothing -- yet the operator is not a stabilizer, so it has altered the logical state. Silent logical failure, which is what it means for Three-qubit repetition to have distance 1: some weight-1 error is invisible to it.

What each generator measured

Each row is a stabilizer generator, each column a qubit. A cell is marked when the error and the generator disagree on that qubit in a way that costs a sign. The syndrome bit is the parity of the marked cells in that row — so two marks cancel and read 0.

Stabilizer generators of the Three-qubit repetition code against the error ZII, showing which qubits contribute to each syndrome bit and the resulting bit.
Generator	q0	q1	q2	Marks	Bit
Error	Z	·	·	weight 1	—
ZZI	Z	Z	·	0	0
IZZ	·	Z	Z	0	0

Syndrome 00 — every generator commuted with this error, so the measurement returned nothing at all.

Decode and correct

The error applied, the syndrome measured, the correction the decoder inferred, and the operator left on the state afterwards.
Error applied	ZII	weight 1; never observed by the decoder
Syndrome measured	00	the only thing the decoder sees
Correction inferred	III	a different operator with the same syndrome
Residual	ZII	a logical operator, so the encoded state has changed

Success is the residual being a stabilizer, not the decoder naming the error. Those differ: on a degenerate code the decoder routinely names a different operator and restores the state perfectly, and scoring by equality would count that as a failure.

Every error of weight 1 and 2

All Pauli errors of each weight on the Three-qubit repetition code, classified by outcome. Computed exhaustively, not sampled.
Weight	Errors	Corrected	Miscorrected	Undetected	Trivial
1	9	3	3	3	0
2	27	6	18	0	3

“Trivial” counts operators that are themselves stabilizers — they act as the identity, so there is nothing to detect. This code is degenerate: 4 distinct syndromes cover more errors than that, so some single-qubit errors are indistinguishable — and are corrected anyway.

The decoder’s lookup table

The lowest-weight error consistent with each syndrome the Three-qubit repetition code can report. This is minimum-weight decoding done exhaustively, which is honest at this size and is not a substitute for matching on a large code.
Syndrome	Correction applied	Weight
00← measured	III	0
01	IIX	1
10	XII	1
11	IXI	1

Decoder benchmarks measured under identical conditions live in the QEC Lab. This page computes exactly and exhaustively on codes small enough for that to be honest; it is a way to see what a syndrome is, not a decoder benchmark.