ToricCode — Kitaev 2003 Z₂ Surface Code

ToricCode is the canonical topological-order benchmark model: Kitaev's (2003) Z₂ surface code on the square lattice. All physical observables exposed here are closed-form / purely topological — no numerical solver is involved.

Hamiltonian

S = 1/2 on every edge of a square lattice. For each vertex v and each plaquette p define the stabilizer operators

A_v = ∏_{i ∈ star(v)} σˣ_i      (vertex / "electric" stabilizer, 4-body)
B_p = ∏_{i ∈ ∂p}     σᶻ_i      (plaquette / "magnetic" stabilizer, 4-body)

H = − J_e Σ_v A_v − J_m Σ_p B_p,    J_e, J_m ≥ 0.

Each pair of stabilizers shares 0 or 2 edges, hence anti-commutes by an even number of factors and so commutes. The full set {A_v} ∪ {B_p} is mutually commuting, and H is exactly diagonal in their joint eigenbasis. The ground state has every A_v = +1 and every B_p = +1.

Closed-Form Quantities

Anyon Content

Four Abelian anyons with quantum dimension d_a = 1:

Fusion rules: e×e = m×m = ε×ε = 1, e×m = ε. The mutual phase from braiding e once fully around m is π (Z₂ mutual semion); this phase is responsible for the ε self-statistics.

Total quantum dimension 𝒟 = √(Σ_a d_a²) = √4 = 2, matching the Kitaev–Preskill / Levin–Wen extraction γ = log 𝒟 = log 2.

Code Examples

using QAtlas

# Default isotropic point
m = ToricCode()                      # J_e = J_m = 1
QAtlas.fetch(m, GroundStateEnergyDensity(), Infinite())   # -2.0
QAtlas.fetch(m, MassGap(),                  Infinite())   # 2.0
QAtlas.fetch(m, GroundStateDegeneracy(),    PBC(0))       # 4 (torus)
QAtlas.fetch(m, GroundStateDegeneracy(),    PBC(0); genus=2)  # 16
QAtlas.fetch(m, TopologicalEntanglementEntropy(), Infinite()) # log 2

# Anisotropic
m2 = ToricCode(J_e=2.0, J_m=0.5)
QAtlas.fetch(m2, GroundStateEnergyDensity(), Infinite())  # -2.5
QAtlas.fetch(m2, MassGap(),                  Infinite())  # 1.0  (= 2·0.5)

# Anyon table
QAtlas.fetch(m, AnyonStatistics())                     # default :em
QAtlas.fetch(m, AnyonStatistics(); type=:e)            # NamedTuple for e
QAtlas.fetch(m, AnyonStatistics(); type=:ε)            # NamedTuple for ε

Distinction from KitaevHoneycomb

The KitaevHoneycomb model (Kitaev 2006, Annals 321) shares an author and a topological-order theme but is otherwise a different physical system:

References

1. A. Yu. Kitaev, "Fault-tolerant quantum computation by anyons", Annals Phys. 303, 2 (2003).
2. A. Kitaev, J. Preskill, "Topological entanglement entropy", Phys. Rev. Lett. 96, 110404 (2006).
3. M. Levin, X.-G. Wen, "Detecting topological order in a ground state wave function", Phys. Rev. Lett. 96, 110405 (2006).
4. C. Nayak, S. H. Simon, A. Stern, M. Freedman, S. Das Sarma, "Non-Abelian anyons and topological quantum computation", Rev. Mod. Phys. 80, 1083 (2008).

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Verified hubs

In the Verified Atlas, this model registers 5 hubs (quantity / BC pair). The badge column shows the R1 assurance level; click a hub link to see the exact verify(...) calls, references, and corroboration mechanism.

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