We start with the consideration of fusion rules of anyonic particles evolving on a 2D surface and the a hypergroup comes with it to construct entangled quantum Markov chains. The fusion rules induce an association scheme with Krein parameters and their duals the intersection numbers. One useful way to think of the schemes as regular graphs encoding the paths of possible quantum walks (automorphisms). We consider braid B3 that describes the unitary dynamics of the anyons as the automorphism subgroup of the graphs. The dynamics induced by the fusions (and the adjoint splitting operations) may be viewed as the chain evolving on a growing graph and the braiding as automorphisms on a fixed graph. In our quantum probability framework infinite iterations of the unitaries, which can encode algorithmic content for quantum simulations, can describe asymptotics elegantly if the particles are allowed to evolve coherently for a longer period. We will define quantum states on the Bose-Mesner algebra which is also a von Neumann algebra as well as a Frobenius algebra to build the quantum Markov chains providing another perspective to topological computation.
We start with the consideration of fusion rules of anyonic particles evolving on a 2D surface and the a hypergroup comes with it to construct entangled quantum Markov chains. The fusion rules induce an association scheme with Krein parameters and their duals the intersection numbers. One useful way to think of the schemes as regular graphs encoding the paths of possible quantum walks (automorphisms). We consider braid B3 that describes the unitary dynamics of the anyons as the automorphism subgroup of the graphs. The dynamics induced by the fusions (and the adjoint splitting operations) may be viewed as the chain evolving on a growing graph and the braiding as automorphisms on a fixed graph. In our quantum probability framework infinite iterations of the unitaries, which can encode algorithmic content for quantum simulations, can describe asymptotics elegantly if the particles are allowed to evolve coherently for a longer period. We will define quantum states on the Bose-Mesner algebra which is also a von Neumann algebra as well as a Frobenius algebra to build the quantum Markov chains providing another perspective to topological computation.
The fusion basis of Fibonacci anyons supports unitary braid representations that can be utilized for universal quantum computation. We show a mapping between the fusion basis of three Fibonacci anyons, {∣1⟩,∣τ⟩}, and the two length 4 Dyck paths via an isomorphism between the two dimensional braid group representations on the fusion basis and the braid group representation built on the standard (2,2) Young diagrams using the Jones construction. This correspondence helps us construct the fusion basis of the Fibonacci anyons using Dyck paths as the number of standard (N,N) Young tableaux is the Catalan number, CN . We then use the local Fredkin moves to construct a spin chain that contains precisely those Dyck paths that correspond to the Fibonacci fusion basis, as a degenerate set. We show that the system is gapped and examine its stability to random noise thereby establishing its usefulness as a platform for topological quantum computation. Finally, we show braidwords in this rotated space that efficiently enable the execution of any desired single-qubit operation, achieving the desired level of precision(∼10−3).
Models for topological quantum computation are based on braiding and fusing anyons (quasiparticles of fractional statistics) in (2+1)-D. The anyons that can exist in a physical theory are determined by the symmetry group of the Hamiltonian. In the case that the Hamiltonian undergoes spontaneous symmetry breaking of the full symmetry group G to a finite residual gauge group H, particles are given by representations of the quantum double D(H) of the subgroup. The quasi-triangular Hopf Algebra D(H) is obtained from Drinfeld's quantum double construction applied to the algebra F(H) of functions on the finite group H. A major new contribution of this work is a program written in MAGMA to compute the particles (and their properties - including spin) that can exist in a system with an arbitrary finite residual gauge group, in addition to the braiding and fusion rules for those particles. We compute explicitly the fusion rules for two non-abelian group doubles suggested for universal quantum computation: S3 and A5, and discover some interesting results, subsystems, and symmetries in the tables. SO(3)_4 (the restriction of Chern-Simons theory SU(2)4) and its mirror image are discovered as 3-particle subsystems in the 8-particle S3 quantum double. The tables demonstrate that both S3 and A5 anyons are all Majorana, but this is not the case for all finite groups. In the appendices, the quantum doubles for the remaining nonabelian subgroups of SO(3) - S4, A4, and D4 (the second in the infinite family Dn) - are tabulated and analyzed. In addition, the probabilities of obtaining any given fusion product in quantum computation applications are determined and programmed in MAGMA. Throughout, connections to possible experiments are mentioned.
FINDING: Anyon braiding forms the computational substrate for topological quantum computation, with non-Abelian statistics encoded in braid group representations rather than local qubit states. | MATH: Braid group Bn generators σi satisfy σiσi+1σi=σi+1σiσi+1 (Yang–Baxter equation); non-Abelian anyons carry unitary representations U(σi) with U(σi)U(σj)=U(σj)U(σi) for ∣i−j∣≥2. Fibonacci anyons: fusion rule τ×τ=1+τ, Hilbert space dimension grows as Fibonacci sequence Fn, golden ratio ϕ=(1+5)/2≈1.618 appears as asymptotic dimension growth rate. Topological protection arises from degeneracy of ground states on genus-g surfaces: dimension =kg for level-k Chern–Simons theory. | CONNECTION: The Fibonacci anyon fusion space dimension ratio Fn+1/Fn→1.618 — direct geometric harmony link. Braid group relations ar Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
This paper presents a novel framework that integrates artificial intelligence with superalgebraic structures to simulate topological quantum computation (TQC) in systems governed by Lie superalgebras such as osp(1∣2) and sl(2∣1). By combining symbolic algebra, modular tensor category theory, and generative AI models—including variational autoencoders and diffusion models—the proposed system enables accurate, scalable simulation of non-Abelian anyonic braiding and fusion operations within graded symmetry constraints. The framework encodes superalgebraic invariants and topological rules into neural architectures, allowing physically valid and symmetry-preserving generation of braid paths and fusion outcomes. Experimental results demonstrate high fidelity, topological consistency, and robustness against noise, with successful emulation on quantum hardware platforms such as IBM Q and IonQ. This research opens new avenues for fault-tolerant, algebraically informed quantum simulation, circuit synthesis, and adaptive control in next-generation quantum technologies. Keywords: topological quantum computation, superalgebra, Lie superalgebras, modular tensor categories, osp(1∣2), sl(2∣1), anyons, braiding, fusion rules, symbolic AI, generative models, quantum simulation, quantum invariants, diffusion models, fault-tolerant quantum computing
The asymptotic dynamics of discrete quantum Markov chains generated by the most general physically relevant quantum operations is investigated. It is shown that it is confined to an attractor space in which the resulting quantum Markov chain is diagonalizable. A construction procedure of a basis of this attractor space and its associated dual basis of 1-forms is presented. It is applicable whenever a strictly positive quantum state exists which is contracted or left invariant by the generating quantum operation. Moreover, algebraic relations between the attractor space and Kraus operators involved in the definition of a quantum Markov chain are derived. This construction is not only expected to offer significant computational advantages in cases in which the dimension of the Hilbert space is large and the dimension of the attractor space is small, but it also sheds new light onto the relation between the asymptotic dynamics of discrete quantum Markov chains and fixed points of their generating quantum operations. Finally, we show that without any restriction our construction applies to all initial states whose support belongs to the so-called recurrent subspace.