Why braiding protects the qubit
The qubit is not stored in any one particle. It lives in a relation between them, what a pair adds up to, and no probe touching a single anyon can read or disturb it. That is the whole point: ordinary qubits lose their state when anything nearby jostles them, and this one does not, as long as no stray anyon is created and wound into the braid.
Moving anyons around each other is the only way to change the qubit, and what happens depends only on which ones crossed and which way round, not on the exact path, its speed or its wobble. A crossing done badly is still exactly the same crossing. The braid you build below the layer is the program, and every crossing is one exact instruction.
The partner of the last anyon is parked outside the gate region because two anyons close together can quietly merge, and merging is a readout: it would collapse the qubit to one of its two answers. Far apart, the pair cannot tunnel into each other, and the parked partner only balances the charge of the others without taking part. In the four-anyon encoding it is brought in on purpose and braided like the others, which is the same qubit written a second way.
Algebra, representation and error correction
F and R moves
Fusion trees for three τ anyons with total charge τ come in two bases, and the F move relates them. Every other admissible F-symbol equals 1 in this gauge, and the two R phases are the counter-clockwise exchange eigenvalues in the two fusion channels.
Explicit R and F in this convention: Hormozi et al. 2007 [5], Eqs. 5 and 6; the same generators are drawn in Bonesteel et al. 2005 [3]. Trebst et al. 2008 [6] use the complex-conjugate convention, the mirror-image theory. Ribbon relation: Kitaev 2006 [14].
Pentagon and hexagon
Consistency of the F moves on four anyons is the pentagon equation; consistency of R with F is the hexagon equation. With the convention |((ab)e c)d⟩ = Σf [Fabcd]ef |(a (bc)f)d⟩ they read
For τ × τ = 1 + τ the pentagon with unitarity fixes Fττττ up to phases, and the hexagon then leaves exactly two solutions for R, complex conjugates of each other; the choice is the chirality of the theory. The test suite behind this page checks both equations numerically over all label assignments for the F and R above, and checks that a shifted R phase breaks the hexagon.
Kitaev 2006 [14], Appendix E; Nayak et al. 2008 [4], Sec. III; Hormozi et al. 2007 [5]; Trebst et al. 2008 [6], Secs. 2.4 and 2.5.
Braid-group representation and density
B₃ acts on the two-dimensional fusion space through σ₁ = diag(R1, Rτ) and σ₂ = Fσ₁F, which satisfy σ₁σ₂σ₁ = σ₂σ₁σ₂. Freedman, Larsen and Wang proved that for the Jones representation at a fifth root of unity, which is this model, the closure of the image of Bn contains SU of each irreducible sector for n ≥ 3, so the image of B₃ is dense in PU(2) and every single-qubit gate is approximable by a three-strand braid; with more strands the representation is universal for quantum computation. In the four-anyon encoding B₄ acts on the same space with σ₃ = σ₁, so the image is unchanged. With five anyons at charge τ, B₅ acts irreducibly on the five-dimensional space F₅ = 5 through the generators shown in the Model section.
Compiling: brute-force search over braids of up to 46 crossings reaches ε of order 10−3 in operator-norm distance, and Solovay–Kitaev iteration lengthens the braid as |log ε|c with c ≈ 4 [3]; a 44-crossing weave for iX reaches ε ≈ 8.5 × 10−4 and one Solovay–Kitaev step ε ≈ 4.2 × 10−5 [5]; asymptotically optimal synthesis reaches depth O(log 1/ε) with a probabilistically polynomial-time algorithm [16]. The exhaustive compiler above stops at 14 crossings, which is why its errors stay at a few times 10−3.
Freedman, Larsen, Wang 2002 [2, 15]; Bonesteel et al. 2005 [3]; Hormozi et al. 2007 [5]; Kliuchnikov, Bocharov, Svore 2014 [16].
Leakage in two-qubit braids
Braiding an anyon from one qubit through another leaves the computational subspace in general: the pair of triples has a larger fusion space than two qubits. Bonesteel et al. avoid this with an injection weave that moves a pair of anyons as a unit through the second qubit and back, so the braid stays inside a total-charge eigenstate; in the limit of an exact injection weave there is no leakage. Their controlled-NOT has ε ≈ 1.8 × 10−3 and 1.2 × 10−3 for total charge 1 and τ of the six anyons, with the injection weave itself at ε ≈ 1.5 × 10−3 [3]. Hormozi et al. give the general construction and show that leakage can be reduced to zero in the limit ε → 0 [5].
Error correction and the threshold
The Fibonacci Turaev–Viro code, the string-net code whose anyons are Fibonacci anyons, has a code-capacity threshold of 4.7% for fixed-rate sampling depolarizing noise with a clustering decoder and 7.3% for pure dephasing, with perfect measurements [7]. Dauphinais and Poulin proved that a fault-tolerance threshold exists for a class of non-cyclic anyon models with noisy measurements and simulated Ising anyons, finding a threshold between 10−4 and 10−3 [8]; the Fibonacci numbers quoted here come from the Turaev–Viro code numerics.
Encodings
Three τ anyons with total charge τ form the minimal Fibonacci qubit used in the compiling literature [3, 5]; the third state of the triple, with total charge 1, is non-computational and decoupled. Four τ anyons with total charge 1 span the same two-dimensional space (F₃ = 2), the pair basis |((ττ)a (ττ)a)1⟩ coincides with the three-anyon basis, and the charge-neutral form is the one drawn in interferometry and readout schemes [4]. Two anyons at fixed total charge span one state, so a single pair is not a qubit. Five τ anyons at charge τ span F₅ = 5 states: the full fusion space, shown here with all four generators, not a qubit.
Why this is a computer
The qubit is not stored in any one anyon. It is the fusion channel of a pair, a property no local measurement on a single particle can see. A crossing is a topological operation: the unitary depends only on which strands crossed and in which sense, not on the path or its speed. Perturbations that do not create anyons and braid them cannot change it.
The three-strand braid group acts on the two-dimensional fusion space through σ₁ and σ₂. Its image is dense in SU(2) up to a phase, so any single-qubit gate can be approximated by a long enough braid; the compiler above shows the error falling as the word length grows. With more anyons the same mechanism gives entangling gates, which is what makes Fibonacci anyons universal for quantum computation.
Conventions. R phases and F matrix as written in Hormozi et al. (2007), the same generators as in Bonesteel et al. (2005); Trebst et al. (2008) use the complex-conjugate convention, the mirror-image theory. σi is drawn as a right-handed rotation about the time axis, so its inverse is the mirror path. As of October 2026, Fibonacci anyons are predicted, not yet observed, in any material; the ν = 12/5 fractional quantum Hall state (Read–Rezayi k = 3) is the classic candidate, moiré fractional Chern insulators the newer one. They have been created and braided as digitally prepared string-net states on qubit processors, without intrinsic topological protection.