Q04 — Toric Code
| Field | Value |
|---|---|
| Domain | Quantum Computing |
| System | 2D surface code on an L×L torus |
| Group | ℤ₂ × ℤ₂ |
| H^k tier | H¹ |
| ISA | Origami (β → ∞) |
| Status | Validated |
| Opcodes | ORBIT · TWIST |
| Paper | Paper 446 |
Physical system
Kitaev’s toric code places one qubit on each edge of an L×L square lattice with periodic boundary conditions. Vertex operators A_v = ⊗ X and plaquette operators B_p = ⊗ Z generate the stabiliser group. The code encodes 2 logical qubits (one per non-contractible cycle on the torus) with distance d = L.
Target category
QCirc — stabiliser sector of quantum circuits over ℂ².
Interpretation functor
F: C → QCirc defined by:
| Opcode | F(opcode) in QCirc |
|---|---|
| ORBIT | Stabiliser measurement: vertex (A_v) or plaquette (B_p) eigenvalue ±1 |
| TWIST | Logical operator: non-contractible string of X or Z along a cycle of the torus |
ISA programme
INIT: LABEL[|+⟩^⊗edges] -- initialise all edge qubits in |+⟩
STAB: ORBIT[A_v] × ORBIT[B_p] -- measure all vertex and plaquette stabilisers
SYNDR: LABEL[∂e | violated] -- identify error chain endpoints (anyons)
MATCH: ORBIT[min-weight matching] -- pair anyons by shortest path (MWPM decoder)
CORRECT: TWIST[string operator] -- apply X or Z string connecting matched pairs
LOGICAL: TWIST[γ_x or γ_z] -- logical X/Z = non-contractible cycle operator
Computable output
- Syndrome: set of violated stabilisers = anyon positions on the lattice.
- Logical error rate p_L ≈ exp(−αL) for p below threshold p_th ≈ 10.3%.
- H¹ structure: the two logical qubits correspond exactly to the two generators of H¹(T², ℤ₂) — the non-contractible cycles of the torus. The TWIST opcode is literally a 1-cocycle. Error correction = finding the homologically trivial completion of an error chain. This is why the toric code is the canonical H¹ QEC example in Paper 446.
- No H² needed: anyons are abelian (ℤ₂ × ℤ₂ fusion); non-abelian anyons (Fibonacci, Ising) require H² BIND. The toric code sits squarely in H¹.
Validation
- Threshold p_th ≈ 10.3% confirmed by mapping to 2D random-bond Ising model (Nishimori point).
- H¹(T², ℤ₂) = ℤ₂ ⊕ ℤ₂: exactly 2 logical qubits on the torus (0 on the sphere, confirming H¹ = homology of the surface).
- BKT connection: Paper 446 identifies the toric code threshold as the β = 1/2 snap event in the XXZ β-ladder.