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International Journal of Academic and Industrial Research Innovations(IJAIRI)

AI-Enhanced Simulation of Superalgebra-Based Topological Quantum Computation

Murali Krishna Pasupuleti

Published Apr 11, 2025• 0 citations
Publisher

Abstract

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

DOI
10.62311/nesx/rp0725

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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
We propose a dual-architecture quantum simulation framework for modeling morphisms and stability conditions in the bounded derived category Db(Coh(X))\mathbf{D}^b(\mathrm{Coh}(X))Db(Coh(X)), with applications to D-brane physics on Kähler and non-Kähler manifolds. Two physically executable quantum realizations are constructed: parameterized quantum circuits (PQCs) implemented on conventional gate-based qubit platforms, and a topological quantum computing (TQC) realization using braiding and fusion of Fibonacci anyons modeled via SU(2)3_33​ modular tensor categories. In the PQC model, we encode slope functionals S(F) and stability constraints as variational observables, mapping derived morphisms to unitaries that evolve over parameterized angles. The output expectation values simulate quantum-corrected Chern class inequalities with deformation terms δδδ, capturing quantum corrections to classical geometric stability. In the TQC model, we engineer braid group representations to implement functorial transformations such as spherical twists and autoequivalences as sequences of fault-tolerant braid operations. This bifurcated approach provides a robust engineering pipeline for simulating categorical stability and homological algebra on quantum hardware, bridging abstract derived category theory with executable quantum architectures.
We provide a comprehensive systematic method for the numerical computation of elementary braid operations in topological quantum computation (TQC). This {procedure} is systematically applicable to all anyon models, including SU(2)kSU(2)_kSU(2)k​. Braiding non-abelian anyons is the essence of TQC, offering a topologically protected implementation of quantum gates. However, obtaining elementary braid matrix representations starting from the fusion and rotation matrices of a specific anyon model is {theoretically guarenteed but no numerical method is given, especially for systems with numerous anyons and complex fusion patterns. Our proposed method addresses this challenge, first in the special case of sparse encoding, allowing for the inclusion of an arbitrary number of anyons per qudit, {and in the general case}. This is accomplished by introducing two methods, one is based on a novel braiding move we call knitting, {the second introduces more general algorithm which is optimal in number of required moves}. The method plays a key role in a broad topological quantum circuit simulator, enabling the examination and study of complex quantum circuits within the TQC framework. Importantly, it proves effective across various anyonic models, accommodating diverse fusion rules. We validate the method by simulating an approximated CNOT gate and present a first-of-a-kind GHZ state simulation on five qubits using three Fibonacci anyons per qubit.
Current quantum computing platforms, primarily based on Z2-graded qubits, suffer from fragility against decoherence and limited error correction thresholds. Here, we propose a topological quantum computing architecture founded on the finite-dimensional 19-dimensional Z3-graded Lie superalgebra and its emergent discrete 44-vector vacuum lattice—a minimal, closed geometric realization of ternary symmetry in 3D embedding space. The lattice supports stable non-Abelian anyonic excitations encoded as native qutrits, with triality-protected braiding offering intrinsic topological error correction and enhanced coherence times. We derive universal gate operations from graded bracket closure, estimate fault-tolerance thresholds exceeding 1.5% noise (significantly surpassing conventional surface code thresholds of ∼0.7–1%), and outline near-term experimental pathways in photonic lattices, cold atoms, and superconducting circuits. This Z3 framework provides a promising candidate for scalable, decoherenceresistant quantum computation, potentially resolving current bottlenecks in qubit-based platforms while bridging algebraic unification with practical quantum hardware.
FINDING: Topological quantum computing uses non-Abelian anyons (Majorana zero modes) for fault-tolerant qubits, with braiding operations encoding quantum gates. | MATH: Braid group Bn B_n Bn​ generators satisfy σiσi+1σi=σi+1σiσi+1 \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1} σi​σi+1​σi​=σi+1​σi​σi+1​; Majorana operators γi \gamma_i γi​ satisfy {γi,γj}=2δij \{\gamma_i, \gamma_j\} = 2\delta_{ij} {γi​,γj​}=2δij​; fusion rules for Ising anyons: σ×σ=1+ψ \sigma \times \sigma = 1 + \psi σ×σ=1+ψ, σ×ψ=σ \sigma \times \psi = \sigma σ×ψ=σ, ψ×ψ=1 \psi \times \psi = 1 ψ×ψ=1. | CONNECTION: Braid group is deeply linked to the golden ratio via the Temperley-Lieb algebra at q=eiπ/5 q = e^{i\pi/5} q=eiπ/5, where the Jones polynomial yields τ=(1+5)/2≈1.618 \tau = (1+\sqrt{5})/2 \approx 1.618 τ=(1+5​)/2≈1.618; Fibonacci anyons have fusion rules τ×τ=1+τ \tau \times \tau = 1 + \tau τ×τ=1+τ, directly encoding the golden ratio. | DEPTH: 9 — Non-Abelian statistics and braiding are foundational to topological quantum computation, with direct geometric harmony in Fibonacci anyon models. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Topological quantum computing uses anyons with non-Abelian braiding statistics for fault-tolerant qubits, with Microsoft's Majorana 1 chip claiming topological qubit realization. | MATH: Braiding group Bn B_n Bn​ generators satisfy σiσi+1σi=σi+1σiσi+1 \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1} σi​σi+1​σi​=σi+1​σi​σi+1​; anyon fusion rules a×b=∑cNabcc a \times b = \sum_c N_{ab}^c c a×b=∑c​Nabc​c; Fibonacci anyon quantum dimension d=ϕ=(1+5)/2≈1.618 d = \phi = (1+\sqrt{5})/2 \approx 1.618 d=ϕ=(1+5​)/2≈1.618; Majorana zero modes obey γi†=γi,{γi,γj}=2δij \gamma_i^\dagger = \gamma_i, \{\gamma_i, \gamma_j\} = 2\delta_{ij} γi†​=γi​,{γi​,γj​}=2δij​. | CONNECTION: Fibonacci anyon dimension d=1.618 d = 1.618 d=1.618 is the golden ratio; braid group representations relate to Temperley-Lieb algebras with parameter τ=1/d2=0.382 \tau = 1/d^2 = 0.382 τ=1/d2=0.382; anyon fusion rules mirror SU(2)_k modular tensor categories with q=e2πi/(k+2) q = e^{2\pi i/(k+2)} q=e2πi/(k+2), linking to root systems An,Dn,E6,E7,E8 A_n, D_n, E_6, E_7, E_8 An​,Dn​,E6​,E7​,E8​. | DEPTH: 8 — Directly ties golden ratio, modular tensor categories, and non-Abelian statistics to fault-tolerant quant Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Topological quantum computation aims to encode quantum information in the nonlocal degrees of freedom of non-Abelian anyons and manipulate it through braiding and fusion operations， offering a promising pathway to fault-tolerant quantum computation. Non-Abelian anyons， with their nontrivial braiding statistics and intrinsic topological fault tolerance， are regarded as a key building block for realizing scalable and decoherence-resistant quantum computing. This review systematically surveys the key developments in anyon physics over the past four decades， from the early theoretical prediction of fractional statistics to direct experimental verification of anyon braiding in fractional quantum Hall states. The physical principles and key findings of essential experimental schemes， including Fabry-Pérot interferometers， Mach-Zehnder interferometers， and anyon colliders， are comprehensively covered. Based on these foundations， the characterization of topological order in even-denominator fractional quantum Hall states， exemplified by the ν=5/2 state， and transport and interferometric probes of non-Abelian statistics are discussed in detail. Furthermore， recent advances in topological quantum states in novel material platforms， including graphene and twisted transition metal dichalcogenides， are reviewed， together with artificial quantum platforms that simulate non-Abelian braiding using superconducting quantum processors. Finally， the key challenges and future directions on the path from anyon physics to scalable topological quantum computing prototype devices are outlined.