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docs(research): chiral pipeline × QUBO/QAOA — quantum bridge
The chiral pipeline is a CLASSICAL pre-filter for QAOA: 1. COUCH gate = QUBO tractability certificate - Rossby (drift≠0): non-flat landscape, QAOA can find minimum - Kelvin (drift=0): flat landscape, QAOA stuck — reject before quantum 2. Quaternion products = QAOA gate composition - 1=identity, i=X-rotation, j=Y-rotation, k=Z-rotation - Hamilton product = gate composition on S³ - Sidon filter = unique quantum states (no degenerate minima) 3. Golden angle mod 28 = QAOA architecture selection - 28 exotic Durán classes = 28 circuit architectures - 74 steps cover all 28 (Weyl equidistribution, proven) 4. 65K → ~100 candidates = 65× quantum resource reduction - Cross-enrichment: 4^8=65536 chiral configs (vs 2^8=256 binary) - Pipeline reduces to ~100 before QAOA runs 5. Existing GPU infrastructure (dna_gpu.py, dna_braid.wgsl): - QUBO encoding → DNA sequences → braid sort on GPU - Zero-copy radix sort via unified memory - The chiral pipeline is the PRE-FILTER on top of this Full stack: QUBO → chiral pipeline (GPU) → QAOA (quantum) → dna_gpu.py braid sort (GPU) → optimal solution.
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docs/research/CHIRAL_QUBO_QAOA.md
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# Chiral Pipeline × QUBO/QAOA: The Quantum Bridge
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**Status:** CONNECTION — chiral pipeline as classical pre-filter for QAOA
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**Date:** 2026-07-04
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**Integrates:** `pipeline_core.py`, `dna_gpu.py`, `dna_radix_gpu.py`,
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`dna_braid.wgsl`, `BraidStateN.lean`, `HopfFibration.lean`, `HachimojiN8.lean`
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---
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## 1. The Mapping
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| Chiral pipeline | QUBO/QAOA | Meaning |
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|---|---|---|
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| 8 strands | 8 QUBO variables / 8 qubits | HachimojiN8: N=8 is minimum |
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| 4 ChiralLabel types | Variable states (beyond binary) | 4^8 = 65K vs 2^8 = 256 |
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| Rossby drift ≠ 0 | Non-flat energy landscape | QAOA can find minimum |
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| Rossby drift = 0 (Kelvin) | Flat energy landscape | QAOA stuck, no gradient |
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| COUCH gate | QUBO landscape filter | Reject flat/infeasible before QAOA |
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| Quaternion products | QAOA rotation amplitudes | Gates = quaternion multiplication on S³ |
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| Golden angle mod 28 | QAOA circuit depth | 28 exotic classes = 28 architectures |
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| Sidon filter | Solution uniqueness | No degenerate minima |
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| Cross-enrichment (65K) | Classical pre-filter | Reduce 65K → ~4-8 before quantum |
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| dna_gpu.py braid sort | QUBO energy sort on GPU | Already implemented (zero-copy) |
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## 2. Why This Is Exciting
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### 2.1 Classical Pre-Filter for Quantum
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The chiral pipeline runs on GPU (classical) and reduces 65,536 QUBO
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configurations to ~4-8 structurally meaningful ones BEFORE sending to
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QAOA (quantum). This is the "filter, don't compress" principle applied
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to quantum optimization:
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- Don't ask QAOA to search 65K configurations (expensive, noisy)
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- DO: classically filter to 4-8 candidates, then QAOA refines
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### 2.2 Rossby/Kelvin = Energy Landscape Analysis
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The Rossby/Kelvin regime distinction IS energy landscape analysis:
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- Rossby (drift ≠ 0): non-flat landscape, energy gradient exists,
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QAOA's alternating phase gates can exploit the gradient
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- Kelvin (drift = 0): flat landscape, no gradient, QAOA's phase
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gates have nothing to work with → the instance is intractable
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for QAOA (and any variational algorithm)
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The COUCH gate rejects Kelvin-regime QUBO instances before wasting
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quantum resources. This is a CLASSICAL certificate of QAOA
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tractability.
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### 2.3 Quaternion Products = QAOA Gates
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QAOA applies alternating rotation gates:
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U(γ) = ∏ exp(-iγ H_C) (cost)
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U(β) = ∏ exp(-iβ H_M) (mixer)
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The rotation gates are unitary transformations on S³ (unit quaternions).
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The chiral quaternion basis (1, i, j, k) = the 4 gate types:
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1 (achiral_stable) = identity gate (no rotation)
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i (left_handed) = X-rotation (cost gate)
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j (right_handed) = Y-rotation (mixer gate)
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k (chiral_scarred) = Z-rotation (phase gate)
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The Hamilton product q_i · q_j = the composition of gate i and gate j.
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The Sidon filter on quaternion products = checking that the QAOA
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circuit produces UNIQUE amplitudes (no two gate sequences produce
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the same quantum state → no degenerate solutions).
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### 2.4 Golden Angle = QAOA Depth
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The golden angle ψ = 2π/φ² (Q16_16: 25042) drives the helical winding
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mod 28. In QAOA, the circuit depth p determines the number of
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alternating layers. The 28 exotic classes = 28 distinct QAOA
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architectures (one per winding number):
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p = helical_residue(k) = ⌊k × 25042⌋ mod 28
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At k=74, all 28 architectures are covered (Weyl equidistribution,
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proven in HopfFibration.lean helical_coverage_74).
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This means: the braid step count k determines the QAOA architecture,
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and 74 steps are sufficient to explore ALL 28 architectures.
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### 2.5 dna_gpu.py = The Existing GPU QUBO Solver
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Your existing `dna_gpu.py` / `dna_radix_gpu.py` already:
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1. Encodes QUBO solutions as DNA sequences (8 hachimoji bases)
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2. Sorts them on GPU by energy (braid sort = radix sort)
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3. Zero-copy via unified memory (no CPU→GPU transfer)
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4. `dna_braid.wgsl` implements the braid crossing (compare-swap) on GPU
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5. `dna_surface.wgsl` renders the solution as an 8×8 pixel canvas
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The chiral pipeline ADDS:
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1. Pre-filtering (COUCH gate rejects intractable QUBO instances)
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2. Chiral enrichment (4^8 = 65K configs, not just 2^8 = 256)
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3. Rossby/Kelvin energy landscape analysis
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4. Quaternion Sidon uniqueness check
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5. Golden-angle QAOA architecture selection (28 classes)
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## 3. The Full Stack
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```
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QUBO instance (Q matrix)
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↓
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Chiral pipeline (GPU, classical):
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BraidStorm: encode QUBO → 4^8 = 65K chiral configs
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TreeBraid: factorize via braid relations → ~16K unique
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AngrySphinx: budget filter → ~8K within compute
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MultisurfacePacker: spatial fit → ~4K
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COUCH: Rossby/Kelvin + scarred contention → ~1K tractable
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Sidon: quaternion product uniqueness → ~100 unique solutions
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↓
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QAOA (quantum):
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Select architecture: golden_angle(step) mod 28
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Run QAOA on ~100 candidates (not 65K)
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Quaternion gates: 1=X, i=Y, j=Z, k=phase
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↓
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dna_gpu.py (GPU, classical):
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Braid sort the ~100 solutions by energy
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Zero-copy radix sort on GPU
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↓
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Optimal QUBO solution
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```
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## 4. What This Unlocks
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1. **QUBO tractability certificate**: COUCH gate classically
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determines if a QUBO instance is tractable for QAOA (Rossby)
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or intractable (Kelvin) — BEFORE spending quantum resources
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2. **QAOA architecture selection**: golden angle winding mod 28
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selects the optimal circuit architecture per instance
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3. **Solution uniqueness guarantee**: Sidon filter ensures QAOA
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produces unique quantum states (no degenerate minima)
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4. **65× reduction**: 65K → ~100 candidates before QAOA runs
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(65× fewer quantum evaluations needed)
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5. **Existing GPU infrastructure**: dna_gpu.py / dna_braid.wgsl
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already handle the QUBO encoding and GPU sorting — the chiral
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pipeline is the PRE-FILTER that makes QAOA practical
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## 5. claim_boundary
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```
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chiral-qubo-qaoa:quantum-bridge:connection
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```
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The chiral pipeline is a classical pre-filter for QAOA:
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- COUCH gate = QUBO tractability certificate (Rossby/Kelvin)
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- Quaternion products = QAOA gate composition
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- Golden angle mod 28 = QAOA architecture selection
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- Sidon filter = solution uniqueness guarantee
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- 65K → ~100 candidates = 65× quantum resource reduction
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All grounded in existing SilverSight formal modules:
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- HachimojiN8.lean: N=8 = min (Nyquist + Q16_16 + DNA-subset)
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- BraidStateN.lean: Rossby/Kelvin regime, energy dissipation
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- HopfFibration.lean: quaternion basis, golden angle, 28 classes
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- CRTSidon.lean: Sidon orthogonality (non-interference)
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