Q03 — Steane 7-qubit Code
| Field | Value |
|---|---|
| Domain | Quantum Computing |
| System | 7-qubit Hamming QEC |
| Group | PSL(2,7) ≅ GL(3,𝔽₂) |
| H^k tier | H¹ |
| ISA | Origami (β → ∞) |
| Status | Validated |
| Opcodes | ORBIT · TWIST · BIND |
| Paper | Paper 325 |
Physical system
The Steane [[7,1,3]] code encodes 1 logical qubit into 7 physical qubits, correcting any single-qubit error. Its stabiliser group is generated by the rows of the parity-check matrix of the [7,4,3] Hamming code, and its transversal gate set includes the full Clifford group — making it the smallest code with a transversal H and CNOT.
Target category
QCirc — the dagger-compact PROP of stabiliser circuits over ℂ².
Interpretation functor
F: C → QCirc defined by:
| Opcode | F(opcode) in QCirc |
|---|---|
| ORBIT | Stabiliser syndrome measurement; eigenvalue = error syndrome s ∈ 𝔽₂³ |
| TWIST | Transversal CNOT / Hadamard; Berry phase = logical phase on encoded qubit |
| BIND | Fano plane incidence constraint; ties 3 stabilisers into a closed H² cycle |
ISA programme
ENCODE: ORBIT[|ψ_L⟩] -- encode logical |ψ⟩ into 7-qubit codeword
SYNDR: ORBIT[s = Hₛe] -- measure 6 syndrome bits (3 X, 3 Z)
LOOKUP: LABEL[e | s] -- identify error location from syndrome table
CORRECT: TWIST[Xᵢ or Zᵢ] -- apply correction on qubit i
BIND: BIND[φ_{ijk}] -- Fano 3-form closes the H² error cycle
DECODE: SPLAT[|ψ⟩] -- recover logical qubit
Code parameters: [[7,1,3]] — distance 3, corrects t=1 error.
Computable output
- Syndrome s ∈ 𝔽₂³: 3-bit string identifying the error location (0 = no error, 1–7 = qubit index in Fano labelling).
- Logical error rate p_L ≈ 21p² for physical error rate p ≪ 1/3.
- H² structure: The Fano plane PG(2,2) is the incidence geometry of the stabiliser generators. Its 7 points and 7 lines correspond to qubits and stabilisers. The BIND opcode captures the non-trivial H² class of this geometry — topologically, Steane is the minimal code where H² is non-zero, which is why it is the first code with a transversal T-gate (via the H² → magic resource connection in Paper 595).
G₂ snap threshold
The Steane code sits at the β = 1/7 G₂ snap point (Paper 572, Zoo entry G2-QEC). When the quantum dimension [7]_β = sin(7πβ)/sin(πβ) = 0, the full Fano orbit loop evaluates to zero: a weight-7 logical error wrapping the entire Fano plane becomes degenerate (invisible). This is the topological protection threshold in G₂ spider language.
The transversal gate group PSL(2,7) ≅ GL(3,𝔽₂) is exactly the discrete symmetry that crystallises from the continuous G₂ at the β=1/7 snap — the snap IS the threshold, and PSL(2,7) IS the surviving symmetry (Zoo entry G2-SSM). This explains why the Steane code has a transversal T-gate: T implements an element of order 7 in PSL(2,7), which becomes available precisely at the snap.
Validation
- Syndrome lookup table exact for all 21 single-qubit Pauli errors (x3 × 7 qubits).
- Fano incidence: each stabiliser generator is a line in PG(2,2); BIND closure = ∂²=0 condition confirmed analytically in Paper 571 (x571a, SHA 214297a).
- Bredon χ cascade: χ_Bredon = 0 at 3-qubit level → Steane is the topological jump (Paper 595).
- G₂ snap at β=1/7: predicted threshold from Paper 572 β-deformation; PSL(2,7) transversal group identified as snap crystallisation (Paper 572 §7, Zoo G2-SSM).