We demonstrate that the two inequivalent spinor representations of the braid group , describing the exchanges of 2 n +2 non-Abelian Ising anyons in the Pfaffian topological quantum computer, are equivalent from the computational point of view, i.e., the sets of topologically protected quantum gates that could be implemented in both cases by braiding exactly coincide. We give the explicit matrices generating almost all braidings in the spinor representations of the 2 n +2 Ising anyons, as well as important recurrence relations. Our detailed analysis allows us to understand better the physical difference between the two inequivalent representations and to propose a process that could determine the type of representation for any concrete physical realization of the Pfaffian quantum computer.
We demonstrate that the two inequivalent spinor representations of the braid group , describing the exchanges of 2 n +2 non-Abelian Ising anyons in the Pfaffian topological quantum computer, are equivalent from the computational point of view, i.e., the sets of topologically protected quantum gates that could be implemented in both cases by braiding exactly coincide. We give the explicit matrices generating almost all braidings in the spinor representations of the 2 n +2 Ising anyons, as well as important recurrence relations. Our detailed analysis allows us to understand better the physical difference between the two inequivalent representations and to propose a process that could determine the type of representation for any concrete physical realization of the Pfaffian quantum computer.
We give a rigorous and self-consistent derivation of the elementary braid matrices representing the exchanges of adjacent Ising anyons in the two inequivalent representations of the Pfaffian quantum Hall states with even and odd numbers of Majorana fermions. To this end we use the distinct operator product expansions of the chiral spin fields in the Neveu–Schwarz and Ramond sectors of the two-dimensional Ising conformal field theory. We find recursive relations for the generators of the irreducible representations of the braid group in terms of those for , as well as explicit formulae for almost all braid matrices for exchanges of Ising anyons. Finally we prove that the braid-group representations obtained from the multi-anyon Pfaffian wavefunctions are completely equivalent to the spinor representations of SO(2 n + 2) and give the equivalence matrices explicitly. This actually proves that the correlation functions of 2 n chiral Ising spin fields σ do indeed realize one of the two inequivalent spinor representations of the rotation group SO(2 n ) as conjectured by Nayak and Wilczek.
We extend the topological quantum computation scheme using the Pfaffian quantum Hall state, which has been recently proposed by Das Sarma et al., in a way that might potentially allow for the topologically protected construction of a universal set of quantum gates. We construct, for the first time, a topologically protected controlled-NOT gate, which is entirely based on quasihole braidings of Pfaffian qubits. All single-qubit gates, except for the $\ensuremath{\pi}∕8$ gate, are also explicitly implemented by quasihole braidings. Instead of the $\ensuremath{\pi}∕8$ gate we try to construct a topologically protected Toffoli gate, in terms of the controlled-phase gate and CNOT or by a braid-group-based controlled-controlled-Z precursor. We also give a topologically protected realization of the Bravyi-Kitaev two-qubit gate g3.
We extend the topological quantum computation scheme using the Pfaffian quantum Hall state, which has been recently proposed by Das Sarma et al., in a way that might potentially allow for the topologically protected construction of a universal set of quantum gates. We construct, for the first time, a topologically protected Controlled-NOT gate which is entirely based on quasihole braidings of Pfaffian qubits. All single-qubit gates, except for the pi/8 gate, are also explicitly implemented by quasihole braidings. Instead of the pi/8 gate we try to construct a topologically protected Toffoli gate, in terms of the Controlled-phase gate and CNOT or by a braid-group based Controlled-Controlled-Z precursor. We also give a topologically protected realization of the Bravyi-Kitaev two-qubit gate g_3.
FINDING: Anyon braiding provides non-Abelian statistics for topological quantum computation, with braid group representations forming the computational basis. | MATH: Braid group Bn generators σi satisfy σiσi+1σi=σi+1σiσi+1 (Yang–Baxter equation); non-Abelian anyons carry unitary braid matrices U(σi) with U(σi)U(σi+1)U(σi)=U(σi+1)U(σi)U(σi+1); Fibonacci anyons have fusion rule τ×τ=1+τ, yielding quantum dimension ϕ=(1+5)/2≈1.618 (golden ratio); topological protection arises from ground-state degeneracy on genus-g surfaces: dimH=4g for toric code, or 2g for Ising anyons. | CONNECTION: Fibonacci anyons' quantum dimension is *exactly* the golden ratio ϕ=1.618, and its inverse ϕ−1=0.618 appears in braid fusion probabilities; the Yang–Baxter equation is a braided analogue of th Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
We give a general proof for the existence and realizability of Clifford gates in the Ising topological quantum computer. We show that all quantum gates that can be implemented by braiding of Ising anyons are Clifford gates. We find that the braiding gates for two qubits exhaust the entire two-qubit Clifford group. Analyzing the structure of the Clifford group for $n\ensuremath{\ge}3$ qubits we prove that the image of the braid group is a nontrivial subgroup of the Clifford group so that not all Clifford gates could be implemented by braiding in the Ising topological quantum computation scheme. We also point out which Clifford gates cannot in general be realized by braiding.
A central question in the theory of topological order asks whether the braid group representation BN carried by N non-Abelian anyons uniquely determines the modular data — the S and T matrices — of the underlying topological quantum field theory. We reconcile three lines of analysis into a single assessment. Structurally, we formalize the question as a reduction map Φ from modular tensor category data to projective braid group representations, whose kernel is the locus of potential counterexamples. Two rigorous finiteness results constrain the problem: braid representations from twisted quantum doubles of finite groups factor through finite groups, and the SO(N)2/O(N)2 braid representations are Gaussian with finite image. We then perform an explicit computation on the Ising anyon model — the canonical non-Abelian order of Majorana zero modes — reconstructing the topological spins θσ=eiπ/8, θψ=−1, the total quantum dimension D=2, and the full S matrix from the braid eigenvalues, with all arithmetic shown. We exhibit a Galois-conjugate deformation Ising(7) with identical fusion rules but distinct S matrix, and verify that its braid representation is inequivalent to the Ising representation on B3 by an explicit trace computation, while remaining indistinguishable at the level of fusion rules. We conclude that braid representations constrain but do not provably uniquely determine modular data in general: uniqueness can hold only under additional hypotheses, and the Majorana fusion-rule classification is not closed by braid data alone. The affirmative claim that braiding is a complete invariant is restricted to a narrower regime than sometimes asserted, and we document this divergence explicitly. ## 1. Introduction Non-Abelian anyons in two spatial dimensions are characterized by two layers of algebraic structure. The first is combinatorial: fusion rules Nabc specifying how anyon c Full text and updates: papers.qnfo.org/papers/braid-group-representations-and-modular-data-non-uniqueness-finite-images-and-th/