Exact Braid Algebra and Diamond-Norm Bounds for a Phenomenological Noisy Fibonacci-Anyon Qubit

This study develops a finite-dimensional, reproducible framework for Fibonacci-anyon quantum computation, combining exact braid algebra with analytical bounds on quantum noise. Three Fibonacci anyons with total topological charge τ encode a two-dimensional logical qubit through the fusion rule τ ⊗ τ = 1 ⊕ τ. Using the golden ratio φ = (1 + √5)/2, the standard fusion matrix F and braiding phases R define the generators B₁ = R and B₂ = FRF, satisfying the Yang–Baxter relation B₁B₂B₁ = B₂B₁B₂. An explicit polynomial certificate reduces this braid identity to two scalar conditions, providing an algebraically transparent and reproducible verification method. The central quantitative result concerns a phenomenological quantum-noise model combining stochastic logical phase flips with orthogonally flagged leakage from the computational fusion space. For phase-flip probability q, leakage probability ℓ, and an arbitrary ideal braid unitary U, the study derives the exact diamond-norm distance between the noisy and ideal quantum channels: ‖𝓔 − 𝓥ᵤ‖◇ = 2[ℓ + (1 − ℓ)q]. The analysis extends to arbitrary stochastic Pauli errors with flagged erasure, yielding ‖𝓔 − 𝓥ᵤ‖◇ = 2[1 − (1 − ℓ)p₀], where p₀ is the identity-Pauli probability. A telescoping composition theorem further establishes circuit-level upper bounds for compatible quantum channels, connecting individual gate errors to cumulative computational accuracy. The work brings together exact symbolic algebra, finite-dimensional quantum-channel analysis, Python-based reproducibility checks, and partial Lean formalization. Its contribution lies in the explicit algebraic certification and closed-form error characterization of a clearly specified effective model, rather than in claiming a new braiding mechanism or experimentally demonstrated fault tolerance. The accompanying Lean sources encode important algebraic and compositional results, while the physical complex-phase specialization and concrete diamond-norm theorems remain outside completed machine-checked verification. Future research can extend the formal proofs, develop fully composable leakage dynamics, incorporate microscopic noise mechanisms, and connect the analytical bounds with experimentally calibrated topological quantum devices.