Algorithm guide · qec
QEC: Steane Seven-Qubit Code
Correct any one-qubit X, Z, or Y error with the [[7,1,3]] CSS code and six stabilizer checks.
Circuit size: 7 data + 1 recycled syndrome ancilla
Repetition-code syndromes · Commuting stabilizers · Binary parity-check columns
Review quantum fundamentals →Notation used in this lesson
- |ψ⟩ and ⟨ψ|
- A state vector (ket) and its conjugate transpose (bra). ⟨a|b⟩ is their inner product.
- α, β and i
- α and β are complex amplitudes; i²=−1. A probability is the squared magnitude |α|².
- XOR, · and ⊗
- XOR adds bits modulo two. In s·x, multiply corresponding bits and XOR them. ⊗ combines registers or operations on different wires.
- U†, Σ and Tr
- U† is the adjoint (inverse for a unitary). Σ sums terms. Tr is the matrix trace, the sum of diagonal entries.
- Probability, shots and angles
- Ideal probabilities describe the model; measured shot frequencies fluctuate. Angles inside trigonometric expressions are in radians unless explicitly labeled turns.
1. The problem
Encode one logical qubit into seven data qubits and correct any single-qubit Pauli X, Z or Y error. Distinguish logical information from the six stabilizer signs used to diagnose the error.
2. How it works
Three Z-type checks locate an X component, and three X-type checks locate a Z component. Each data wire has a distinct three-bit syndrome column. A Y error has both components and activates both check groups.
Encode
Prepare the seven-qubit CSS codeword from the chosen logical state.
Inject
Apply one selected Pauli error on a data qubit.
Extract six checks
Reuse one resettable ancilla sequentially, preserving each result in a distinct classical bit.
Recover and verify
Use syndrome columns for X/Z corrections, decode and undo logical preparation.
3. The mathematics, one idea at a time
Code states have stabilizer eigenvalue +1. If an error anticommutes with g, its check sign changes to −1.
Y_j behaves as both components, up to phase.
These are displayed three-bit integers in its chosen check order; other textbooks may permute the columns.
Work this through by hand
4. Single X error on q0
Seven data plus one recycled ancilla: eight physical qubits. This code has fixed width.
Choose logical |0⟩, error X on q0 and correction on.
| Stage | Calculation or state | What it means |
|---|---|---|
| Which Z checks contain q0? | Z0, Z1 and Z2 | All three anticommute with the injected X. |
| Which X checks flip? | None | X commutes with the X-type checks. |
| Syndrome | c5c4c3 | c2c1c0 = 000|111 | The low check group selects q0 for X recovery. |
| Verification | Seven decoded data-check bits are all zero | The syndrome remains in the six low classical bits. |
Expected outcomes for this example
These are ideal predictions for the stated configuration, not a result already measured by your browser. Rows describing alternative modes or combined events are labeled explicitly.
| Outcome or event | Ideal probability | Interpretation |
|---|---|---|
| 0000000|000|111 | 100% ideal for this configuration | Seven verification bits, three X-check results, three Z-check results. |
5. Load the worked example in Composer
The worked example can use smaller or simpler inputs than the gallery defaults. The button below loads the exact circuit used in this lesson.
- Logical input state
- 0
- Introduce a single-qubit error
- On
- Error type
- X
- Data qubit receiving the error
- 0
- Enable correction
- On
- Save any current circuit you want to keep, then open the worked example. Loading it does not execute a paid hardware job.
- Trace the preparation, main operations and measurements against the worked table. Check the gate editor for controls, targets and angles.
- Start with an ideal noiseless simulator compatible with the circuit. Mid-circuit measurement, reset and feed-forward require support from the chosen backend.
- Inspect the live preview. For a recorded experiment, open Execute, set the available backend options and shots, and Run. Sign in when prompted to retain it in My Jobs.
- Read the output as described below before changing one setting and repeating. The guide describes the original example; edits can change the answer.
6. Read the output correctly
The full output contains 13 classical bits, despite only eight physical qubits, because the ancilla is reused.
Do not compare syndrome bit order with a textbook table until the check convention is matched.
For arbitrary noise, a clean-looking syndrome does not rule out a logical error; single-error correction is the guaranteed setting here.
A bit label is a register value; its bar height is a probability or count. For S shots and observed count k, the sampled frequency is k/S. A typical sampling scale for probability p is √[p(1−p)/S]; this is not a strict error bound. Finite samples do not have to match the ideal percentages exactly.
7. Common mistakes & limits
- [[7,1,3]] means seven physical data qubits, one encoded logical qubit, distance three; it does not mean three-error correction.
- Reset the recycled ancilla between checks. Reusing its old state would change the measurement circuit.
8. Check your understanding
What are the two three-bit syndrome groups for Y on q0?
Reveal the reasoning
Both are 111: the X and Z components each flip their corresponding group.
9. Further study
The lesson’s register mappings, supported inputs and loadable example follow Quantket’s implemented circuit. A reference may use different wire names, bit order or resource conventions.
Gallery reference: defaults, definitions & implementation notes
Inputs & default settings
Choose the logical state |0⟩, |1⟩, |+⟩, |−⟩, |+i⟩ or |−i⟩; whether to introduce an error; its X, Z, Y or seeded-random type; and a data-qubit target or seeded-random target. A seed makes random choices repeatable.
Switch syndrome-conditioned correction on or off.
- logicalState
- Gallery default: +
- protection
- Gallery default: bit
- introduceError
- Gallery default: true
- errorType
- Gallery default: X
- targetQubit
- Gallery default: 0
- correctionEnabled
- Gallery default: true
- seed
- Gallery default: 417
- rounds
- Gallery default: 5
- shots
- Gallery default: 256
- recoveryMode
- Gallery default: pauli-frame
- physicalErrorRate
- Gallery default: 0.002
- readoutErrorRate
- Gallery default: 0.005
Implementation notes
The six checks are measured sequentially with one resettable physical ancilla. The ghost map previews which check outcomes become −1 before the circuit is generated.
Read your saved result
Check the submitted circuit, measured register and execution settings in Run details. Exact results give computed probabilities; sampled results show frequencies from the recorded shots. Edited circuits and noise can change the expected outcome. The algorithm label records the circuit's origin; it does not certify an edited circuit or a measured result.