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…
Braid group as fundamental group: π₁(C_n(R²)) = B_n; Artin presentation with σ_i σ_{i+1} σ_i = σ_{i+1} σ_i σ_{i+1}
Braid group as fundamental group of the unordered configuration space. Setup: n distinguishable (resp. indistinguishable) particles on the…
Topological Quantum Field Theory as cobordism functor: Z: Cob_n → Vect_C (Atiyah-Segal)
Topological Quantum Field Theory via Atiyah-Segal cobordism axioms. Setup: an n-dim TQFT is a symmetric monoidal functor Z: Cob_n → Vect_C…
Chern-Simons action S_CS = (k/4π)∫Tr(A∧dA + (2/3)A∧A∧A); level-k Chern-class quantisation
Chern-Simons topological gauge theory. Setup: a 3-manifold M with a principal G-bundle (G compact semi-simple Lie group, e.g., SU(2));…
Abelian anyon mutual-statistics phase: ψ(a↻b) = exp(2πi·θ)·ψ; Laughlin ν=1/3 gives θ = 1/3 (exp(2πi/3))
Abelian anyon mutual-statistics phase. From framework F7 (braid group π₁), a full braid of particle a around particle b (not an exchange)…
Toric code ground-state degeneracy on genus-g surface: GSD = 4^g (Kitaev 2003)
Toric code ground-state degeneracy on a genus-g closed orientable surface. From framework F8 (TQFT cobordism), the dimension of H_Σ_g is a…
IQHE Hall conductance σ_xy = n·(e²/h), n = first Chern number (TKNN 1982)
Integer quantum Hall effect Hall conductance as Chern number (Thouless-Kohmoto-Nightingale-den Nijs 1982). From framework F9 (Chern-Simons…
Toric code (Kitaev 1997)
A Kitaev 1997 toric-code stabilizer Hamiltonian; topological-protection abelian anyons; basis of surface-code QEC.
Majorana wire (Kitaev 2001)
Kitaev 2001 1D p-wave wire with MZM zero-modes; modern Mourik-Kouwenhoven 2012 ZBP + topological-superconductor experimental signatures.
Non-abelian anyons (Moore-Read 1991)
Moore-Read 1991 Pfaffian-state at nu=5/2; predicted Ising-anyons; modern MZM + Fibonacci-anyon braiding-gates.
Braid group computing
Freedman-Larsen-Wang 2003 universal-quantum-computing via braiding non-abelian anyons; basis of topological-quantum-computing motivation.
Fibonacci anyons
Microsoft Station Q research; nu=12/5 Fibonacci anyons universal for QC; modern Microsoft Majorana 1 chip 2025 announcement.
TI/SC (Fu-Kane 2008)
Fu-Kane 2008 TI/superconductor proximity Majorana-vortices; modern Pb/Bi2Se3 + Bi2Te3 platforms; topological-quantum-computing route.
Toric code (Kitaev 1997)
A Kitaev 1997 toric-code + 2003 fault-tolerant TQC; modern modern foundational text + surface-code Bravyi-Kitaev + 2024 Google Willow chip.
Majorana mode (Mourik 2012)
V Mourik-Kouwenhoven 2012 InSb-NbTiN nanowire; modern modern foundational + 2018 retraction Microsoft Q-Star + 2024 Mzm progress.
Nayak review (2008)
C Nayak-Simon-Stern-Freedman-Sarma 2008 'Non-Abelian Anyons and TQC' RMP; modern modern foundational text + 2024 Microsoft topo-conditional…
Freedman (2003)
M Freedman 2003 + Larsen-Wang topological-quantum-computation; modern modern foundational + 2024 Fibonacci anyons universal-gate-set.
ν=5/2 FQHE (Willett 2009)
R Willett 1987 + 2009 e/4-charge ν=5/2 quasiparticle; modern modern foundational + Pfaffian Moore-Read + 2024 thermal-Hall transport.
Anyons (Leinaas-Myrheim 1977)
J Leinaas-J Myrheim 1977 + Wilczek 1982 anyons; modern modern foundational text + 2020 Bartolomei first anyon-collider + 2024…