Resource Analysis: Scaling & Compilation Overhead
Physical → Encoded → Logical Hierarchy
| Layer | Entity | Scaling |
|---|---|---|
| Physical | τ-anyons | N |
| Encoded | Fusion space | dim ≈ F_{N-1} ≈ φ^N/√5 |
| Logical | Qubits | n ≈ 0.694N - 1.16 |
Asymptotic limit: ~69% qubit density extraction from physical anyons.
SHA-520 Preimage Resources
| Resource | Per-Round | Total (2^260 Grover iterations) | Feasibility |
|---|---|---|---|
| Logical qubits | 2,500 | 2,500 (reused) | NISQ |
| Physical anyons (4-τ) | 10,000 | 10,000 | ~1 cm² area |
| T-gates per iteration | 10⁶ | 2^260 × 10⁶ | IMPOSSIBLE |
| Braid complexity | O(10⁶ × 300) | 3×10⁸ braids/iter | 30 sec/iter |
| Total time | 30 sec | 10^70 years | ❌ IMPOSSIBLE |
Verdict: Quantum computing provides O(2^260) speedup over classical (already optimal via Grover). But 2^260 iterations × 30 sec = 10^70 years = unachievable.
Reduced-Round Analysis
For r-round SHA-520 with truncated b-bit output:
| Rounds | Output bits | Classical | Quantum | Advantage |
|---|---|---|---|---|
| 4 | 16 | 2^16 | 2^8 | ✓ Quadratic |
| 4 | 20 | 2^20 | 2^10 | ✓ Quadratic |
| 8 | 24 | 2^24 | 2^12 | ✓ Quadratic |
| 16 | 32 | 2^32 | 2^16 | ✓ Quadratic |
| 80 (full) | 520 | 2^520 | 2^260 | ✓ Quadratic (useless) |
Key insight: Quantum advantage is real but polynomial (2×). For cryptanalysis, it doesn't matter—still 10^70 years.
Scaling Breakdown
Theorem: Topological advantage is lost at N_crit ≈ 10⁴-10⁵ physical anyons.
Why:
- Adiabatic condition fails: τ_braid ≫ ħ/Δ → Braid time grows with system size
- Control complexity: Need O(N) independent voltage gates for individual anyon control
- Interferometry crosstalk: Measurement visibility decays as exp(-d/ξ) over distance
- Thermal background: Stray anyon density n_th ≈ exp(-Δ/kT) × area increases
- Fabrication: 2DEG uniformity over cm² scale unproven at required precision
Surface code comparison: For small N, surface codes require less overhead (empirically).
Braid Compilation Overhead
Solovay-Kitaev: ε-approximation requires L(ε) = O(log^3.97(1/ε)) braids per T-gate
Practical example:
- Precision ε = 10^-10
- log(1/ε) ≈ 33
- L(10^-10) ≈ 33^3.97 ≈ 1,400,000 braids per T-gate
For 10⁶ T-gates per Grover iteration:
- Total braids per iteration: 1.4 × 10^12
- Time per iteration: 1.4 × 10^12 × 10ns = 14 seconds (much better than above 30s estimate)
Coherence Time Requirements
For full SHA-520 (80 rounds, 2^260 iterations):
t_total ≈ 2^260 iterations × 14 sec/iteration = 10^70 years
T₂ needed > 10^70 years
Topological protection claims: T₂ > 1 second (theoretical) Gap: 10^70 years > 1 second—still impossible.
Error Correction Cycles
Surface code threshold: ~1% physical error → ~10% logical per cycle Topological threshold (conjectured): ~1% (same or better)
No advantage unless:
- ν = 12/5 state exhibits error rates < 0.1% (unproven)
- Adiabatic braiding achieves > 99.5% fidelity (unproven)
- Interferometry visibility > 95% (unproven)
Conclusion
Topological quantum computing cannot break SHA-520 because:
- ✓ Quantum speedup is real (√N for Grover)
- ✗ But scaling goes 2^260 iterations
- ✗ 2^260 × any finite time = impossible
- ✗ Topological advantage (smaller overhead) doesn't matter—still 10^70 years
Same as any other quantum computer for cryptanalysis.
Numbers frozen. No appeals.