G2-QEC — G₂ Snap Events as Quantised QEC Thresholds
| Field | Value |
|---|---|
| Domain | Quantum Error Correction |
| System | Topological codes at Fano-prime qubit count |
| Group | PSL(2,7) at β=1/7; ℤ₅ at β=1/5; ℤ₃ at β=1/3 |
| H^k tier | H² |
| ISA | Origami (β → ∞ topological) / Forge (finite β) |
| Status | Predicted |
| Opcodes | BIND · ORBIT · TWIST |
| Papers | Papers 572, 607 |
Overview
The β-deformed G₂ spider (Paper 572) has snap events at β = 1/7, 1/5, 1/3 where quantum dimensions [7]_β, [5]_β, [3]_β vanish. These are not generic error thresholds — they are root-of-unity quantised thresholds for code families whose stabiliser geometry is based on Fano-prime qubit counts n = 7, 5, 3. The snap is the threshold: at β = 1/p, the p-dimensional anyonic charge of the minimal error path evaluates to zero, making the error chain indistinguishable from the vacuum.
The three QEC snap thresholds
β = 1/3: colour code (weight-3 stabilisers)
What collapses: [3]_β = 0 — the anyonic charge of a weight-3 error chain in the G₂ spider equals zero. A weight-3 error string is indistinguishable from the vacuum: anyon condensation in the weight-3 sector.
Code family: codes whose minimum-weight stabilisers are weight-3 (triangular colour codes). The threshold condition is exactly [3]_β = 0: below β = 1/3 the code corrects minimal errors; at β = 1/3 weight-3 error paths become degenerate with the identity.
Peierls connection (Paper 607 Theorem B): β*(P) = (1/κ)log(1/2B·n_B). For κ = 3 (weight-3 network dimension) and n_B determined by the colour code geometry, Theorem B back-determines the defect fugacity B from the snap condition β* = 1/3.
β = 1/5: five-qubit perfect code [[5,1,3]]
What collapses: [5]_β = 0 — BIND∘BIND† = [5]·id → 0. The error-correction map self-annihilates on composition: applying the correction operator twice gives zero.
Code identity: the [[5,1,3]] perfect code, the smallest quantum code correcting any single-qubit error. It saturates the quantum Hamming bound — and it does so because it operates at β = 1/5. The BIND self-annihilation at [5]=0 is the condition for maximal degeneracy: two distinct errors share the same syndrome, so the code can correct both using one syndrome measurement. Perfection = G₂ snap.
Structural consequence: the 5-qubit code is the only [[n,1,3]] code saturating the Hamming bound because n=5 is the only Fano-prime that admits a [[n,1,3]] perfect code. n=3 gives [[3,1,1]] (trivial repetition); n=7 gives [[7,1,3]] (Steane, not perfect but distance-optimal). The snap at β=1/5 selects n=5 uniquely for perfection.
β = 1/7: Steane [[7,1,3]] code (Fano threshold)
What collapses: [7]_β = 0 — the full Fano orbit loop evaluates to zero. A weight-7 logical operator wrapping the entire Fano plane becomes degenerate (invisible to the code). This is the topological protection threshold in G₂ spider language.
Code identity: the Steane [[7,1,3]] code, encoding 1 logical qubit in 7 physical qubits arranged in Fano-plane geometry (PG(2,2)). Its 6 stabiliser generators are the 6 Fano lines used for syndrome measurement; its transversal gate group is PSL(2,7) ≅ GL(3,𝔽₂) — exactly the discrete symmetry that crystallises from G₂ at the β=1/7 snap (see Zoo entry G2-SSM). The snap IS the threshold; PSL(2,7) IS the surviving symmetry after G₂ breaks.
Why the Steane code has a transversal T-gate: the T-gate generates the third level of the Clifford hierarchy. It is transversal in the Steane code because PSL(2,7) — the gate group that crystallises at β=1/7 — contains elements of order 7 that implement the T-gate exactly. No smaller code has a transversal T because no smaller n is a Fano prime with a G₂ snap.
The QEC cascade
| β snap | Code | n | What collapses at threshold |
|---|---|---|---|
| 1/3 | Colour code | 3 | Weight-3 anyon condensation |
| 1/5 | [[5,1,3]] perfect | 5 | BIND self-annihilation; max degeneracy |
| 1/7 | Steane [[7,1,3]] | 7 | Fano orbit; PSL(2,7) crystallises |
The non-Fano-prime prediction
Codes with n = 4, 6, 8, … (non-Fano-prime) do not sit at G₂ snap points. Their error thresholds are generic: determined by code-specific geometry (Peierls formula Theorem B with code-specific B, κ, n_B), but not quantised at a root-of-unity value.
This is a falsifiable prediction: the thresholds of the 3-, 5-, and 7-qubit codes (under depolarising noise) should be related by simple rational ratios derived from the G₂ quantum dimensions, while the threshold of the 4-qubit or 6-qubit codes should not fit the same pattern.
ISA programme for G₂-quantised threshold detection
INIT: LABEL[n qubits; code geometry] -- identify n
FANO?: LABEL[n in {3,5,7}?] -- check Fano-prime
G2SNAP: BIND[beta* = 1/n] -- assign snap threshold
PEIERLS: LABEL[B from Theorem B consistency] -- back-determine fugacity
PREDICT: LABEL[threshold p* = f(beta*,B,kappa)] -- output threshold
VERIFY: ORBIT[compare to known threshold] -- validate
Connection to fault-tolerant universality
The three snap thresholds give the three critical resources for universal fault-tolerant quantum computation:
| Snap | Code | Resource provided |
|---|---|---|
| β=1/3 | Colour code | Transversal Clifford + T (magic state distillation input) |
| β=1/5 | [[5,1,3]] perfect | Perfect single-error correction; minimal overhead |
| β=1/7 | Steane [[7,1,3]] | Transversal T-gate; CSS structure; full Clifford group |
Together they form the minimal fault-tolerant toolkit. The G₂ snap structure explains why these three codes — and not others — are the canonical primitives of fault-tolerant QC: they are the codes whose stabiliser geometry sits exactly at G₂ snap points.
Part of the ISA Zoo. See also: Q03 Steane code, G2-CHEM snap events, G2-SSM symmetry making, Paper 607.