Physics
Toric code on L×L torus:
H = -∑ Av - ∑ Bp
Av = ∏ Xᵢ, Bp = ∏ Zⱼ
Threshold: p_th ≈ 10.3%
(depolarizing noise)
p < p_th → errors correctable
Logical failure rate:
p_L ~ (p/p_th)^⌈d/2⌉
exponential suppression
Below threshold, increasing code distance d exponentially suppresses logical error rate. Above threshold, larger codes perform worse. This phase transition is equivalent to a 2D random-bond Ising model percolation transition.