# 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.*