Quantum computation using the braiding of non-abelian anyonic quasiparticles in (2+1)-D topologically-ordered matter. Kitaev 2003 fault-tolerant proposal; Nayak-Simon-Stern-Freedman-Das Sarma 2008 review. Anyons are indistinguishable…
topological-quantum-computing
Non-abelian anyons: quasi-particle exchange represented by braid group B_n, dim ≥ 2
In (2+1)D, the configuration space of n indistinguishable particles is not simply-connected; its fundamental group is the braid group B_n…
Kitaev 1D p-wave chain: Majorana zero modes at the ends when |μ| < 2t
Kitaev 2001 toy model: a 1D spinless p-wave paired superconductor. H = Σ_j [−μ c_j†c_j − t(c_j†c_{j+1} + c_{j+1}†c_j) + Δ(c_j c_{j+1} +…
Fibonacci anyons: non-abelian, computationally universal by braiding alone
Fibonacci anyons: the simplest non-abelian anyonic model with computationally universal braid-group representation. Single non-trivial…
Ising-anyon braid unitary: R = exp(iπ/8)·diag(1, i) — single-qubit Clifford
Ising-anyon model: one non-trivial species σ with fusion rule σ ⊗ σ = 1 ⊕ ψ (vacuum + fermion); d_σ = √2. Braid matrix for two σ anyons in…
Kitaev chain topological phase iff |μ| < 2t — gap-closing transition at |μ| = 2t
Bogoliubov-de Gennes spectrum of the Kitaev chain at momentum k: E(k) = ±√((μ + 2t cos k)² + 4Δ² sin² k). Bulk gap closes only when both…
Fibonacci anyon quantum dimension: d_τ = φ (golden ratio); dim(H_n) = F_n
From the fusion rule τ ⊗ τ = 1 ⊕ τ, the quantum dimension d_τ satisfies d_τ² = 1 + d_τ → d_τ = φ = (1+√5)/2 (golden ratio). Equivalent to…