Architecture: Topological Quantum Computer (Fibonacci Anyon Model)

System Overview

PHYSICAL LAYER          LOGICAL LAYER           APPLICATION LAYER
─────────────           ────────────            ─────────────────
2DEG / FQH ν=12/5  ←→  Fusion Space  ←→  Cryptanalytic Algorithm
                        (τ anyons)           (SHA-520 preimage)
                        Braiding Gates
                        (F-moves, R-moves)

1. Fibonacci Anyon Theory (SU(2)₃)

Fusion rules:

τ × τ = 1 + τ
1 × τ = τ
τ × 1 = τ
1 × 1 = 1

Quantum dimensions: d₁ = 1, d_τ = φ = 1.618..., D_total ≈ 1.902

Key theorem: dim(V_n) = F_{n-1} (Fibonacci numbers) for n τ-anyons with total charge 1

2. Braiding (R-Matrices)

Eigenvalues for τ×τ:

R^{ττ}_1 = e^{-4πi/5} (vacuum)
R^{ττ}_τ = e^{3πi/5} (τ channel)

These are 10th roots of unity → dense in SU(2) with F-moves.

3. Logical Qubit Encodings

4-τ Standard (recommended):

|0⟩_L = |((ττ)₁(ττ)₁)₁⟩
|1⟩_L = |((ττ)_τ(ττ)_τ)₁⟩
Total charge = 1 (vacuum) → interferometric measurement possible
4 physical anyons per logical qubit

Asymptotic qubit density: n_max ≈ 0.694N - 1.16 logical qubits from N physical anyons

4. Braid Compilation

Solovay-Kitaev: L(ε) = O(log^3.97(1/ε)) for ε-precision

Pipeline: Clifford+T → Braid word optimization → Solovay-Kitaev → Adiabatic schedule → Voltage gates on 2DEG

5. Scaling Limits

Topological advantage lost at ~10⁴-10⁵ anyons due to:

Adiabatic timing constraints
Control complexity (O(N) gates)
Interferometry crosstalk
Thermal anyon density
Fabrication yield limits

References

Kitaev, A. (2003). "Fault-tolerant quantum computation by anyons." Annals of Physics.
Freedman, Larsen, Wang (2002). "Two-eigenvalue problem and Jones representations."

Frozen by Ahmad. Falsifiable by experiment.