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.

XIIIIa single bit flip ZIIIIa single phase flip YIIIIboth at once XXIIItwo errors at once ZZZZZcommutes with every generator, so nothing is measured XZZXIacts as the identity on the code space

Five-qubit perfect code, error XXIII

Miscorrected — logical error introduced

The syndrome is consistent with the lower-weight error IIIZI, so that is what the decoder applied. The residual is a logical operator, so the correction did not merely fail -- it introduced the logical error the code exists to prevent.

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 Five-qubit perfect code against the error XXIII, showing which qubits contribute to each syndrome bit and the resulting bit.
Generator	q0	q1	q2	q3	q4	Marks	Bit
Error	X	X	·	·	·	weight 2	—
XZZXI	X	[Z]	Z	X	·	1	1
IXZZX	·	X	Z	Z	X	0	0
XIXZZ	X	·	X	Z	Z	0	0
ZXIXZ	[Z]	X	·	X	Z	1	1

Syndrome 1001 — 2 of 4 generators were disturbed.

Decode and correct

The error applied, the syndrome measured, the correction the decoder inferred, and the operator left on the state afterwards.
Error applied	XXIII	weight 2; never observed by the decoder
Syndrome measured	1001	the only thing the decoder sees
Correction inferred	IIIZI	a different operator with the same syndrome
Residual	XXIZI	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 Five-qubit perfect code, classified by outcome. Computed exhaustively, not sampled.
Weight	Errors	Corrected	Miscorrected	Undetected	Trivial
1	15	15	0	0	0
2	90	0	90	0	0

“Trivial” counts operators that are themselves stabilizers — they act as the identity, so there is nothing to detect. Every weight-1 error here has its own syndrome, so the decoder can always name the operator that occurred.

The decoder’s lookup table

The lowest-weight error consistent with each syndrome the Five-qubit perfect 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
0000	IIIII	0
0001	XIIII	1
0010	IIZII	1
0011	IIIIX	1
0100	IIIIZ	1
0101	IZIII	1
0110	IIIXI	1
0111	IIIIY	1
1000	IXIII	1
1001← measured	IIIZI	1
1010	ZIIII	1
1011	YIIII	1
1100	IIXII	1
1101	IYIII	1
1110	IIYII	1
1111	IIIYI	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.