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.

XIIIIIIIIa single bit flip ZIIIIIIIIa single phase flip YIIIIIIIIboth at once XXIIIIIIItwo errors at once ZZZZZZZZZcommutes with every generator, so nothing is measured ZZIIIIIIIacts as the identity on the code space

Shor code, error XXIIIIIII

Miscorrected — logical error introduced

The syndrome is consistent with the lower-weight error IIXIIIIII, 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 Shor code against the error XXIIIIIII, showing which qubits contribute to each syndrome bit and the resulting bit.
Generator	q0	q1	q2	q3	q4	q5	q6	q7	q8	Marks	Bit
Error	X	X	·	·	·	·	·	·	·	weight 2	—
ZZIIIIIII	[Z]	[Z]	·	·	·	·	·	·	·	2	0
IZZIIIIII	·	[Z]	Z	·	·	·	·	·	·	1	1
IIIZZIIII	·	·	·	Z	Z	·	·	·	·	0	0
IIIIZZIII	·	·	·	·	Z	Z	·	·	·	0	0
IIIIIIZZI	·	·	·	·	·	·	Z	Z	·	0	0
IIIIIIIZZ	·	·	·	·	·	·	·	Z	Z	0	0
XXXXXXIII	X	X	X	X	X	X	·	·	·	0	0
IIIXXXXXX	·	·	·	X	X	X	X	X	X	0	0

Syndrome 01000000 — 1 of 8 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	XXIIIIIII	weight 2; never observed by the decoder
Syndrome measured	01000000	the only thing the decoder sees
Correction inferred	IIXIIIIII	a different operator with the same syndrome
Residual	XXXIIIIII	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 Shor code, classified by outcome. Computed exhaustively, not sampled.
Weight	Errors	Corrected	Miscorrected	Undetected	Trivial
1	27	27	0	0	0
2	324	36	279	0	9

“Trivial” counts operators that are themselves stabilizers — they act as the identity, so there is nothing to detect. This code is degenerate: 22 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 Shor 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
00000000	IIIIIIIII	0
00000001	IIIIIIZII	1
00000010	ZIIIIIIII	1
00000011	IIIZIIIII	1
00000100	IIIIIIIIX	1
00000101	IIIIIIIIY	1
00001000	IIIIIIXII	1
00001001	IIIIIIYII	1
00001100	IIIIIIIXI	1
00001101	IIIIIIIYI	1
00010000	IIIIIXIII	1
00010011	IIIIIYIII	1
00100000	IIIXIIIII	1
00100011	IIIYIIIII	1
00110000	IIIIXIIII	1
00110011	IIIIYIIII	1
01000000← measured	IIXIIIIII	1
01000010	IIYIIIIII	1
10000000	XIIIIIIII	1
10000010	YIIIIIIII	1
11000000	IXIIIIIII	1
11000010	IYIIIIIII	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.