QCOS Scientific Validation Summary
Softquantus Inc. — January 2026
Executive Summary
QCOS (Quantum Circuit Optimization System) has been rigorously validated using the H₂ molecule VQE benchmark. All tests pass and results are publication-ready.
Key Results at a Glance
| Validation | Result | Status |
|---|---|---|
| Hamiltonian Equivalence | 16/16 eigenvalues match (diff < 10⁻¹² Ha) | ✅ PASS |
| Shot Noise Scaling | σ² ∝ 1/N with R² = 0.994 | ✅ PASS |
| Unit Test Suite | 35/35 tests pass | ✅ PASS |
| Multi-Start Optimization | 10 restarts optimal (12.7% improvement) | ✅ VALIDATED |
1. Hamiltonian Validation
Claim: Qiskit Nature and OpenFermion produce identical H₂ Hamiltonians.
Evidence:
- All 16 eigenvalues match within machine precision
- Maximum difference: 2.44 × 10⁻¹² mHa
- Trace difference: 3.33 × 10⁻¹⁵
- Physical subspace (2-electron sector): 6/6 eigenvalues match
Conclusion: ✅ SPECTRAL EQUIVALENCE PROVEN
2. Shot Noise Analysis
Claim: Energy estimation variance follows theoretical 1/N scaling.
| Shots | Variance | Error (mHa) |
|---|---|---|
| 512 | 5.58×10⁻⁵ | 0.65 |
| 1,024 | 2.92×10⁻⁵ | 1.15 |
| 4,096 | 6.60×10⁻⁶ | 0.39 |
| 16,384 | 1.94×10⁻⁶ | 0.23 |
Fit: σ² = 0.0286/N − 8.67×10⁻⁸
R² = 0.994 ✅
3. Multi-Start Optimization
Question: How many restarts are optimal for fixed budget?
| Restarts | Budget/Restart | Mean Error (mHa) |
|---|---|---|
| 1 | 100 | 7.71 |
| 2 | 50 | 7.00 |
| 5 | 20 | 6.96 |
| 10 | 10 | 6.73 ← optimal |
Improvement: 12.7% (p = 0.038)
Conclusion: More restarts help exploration for H₂.
4. Test Suite Coverage
| Category | Tests | Description |
|---|---|---|
| Pauli Conventions | 7 | Qubit ordering, Z eigenvalues |
| Measurement Basis | 5 | X/Y rotations, XXYY/YYXX |
| Hamiltonian | 8 | Hermiticity, coefficients, energy |
| Ansatz | 5 | States at θ=0, θ=π, amplitudes |
| Integration | 10 | Statevector vs counts agreement |
| Total | 35 | All pass ✅ |
5. Molecular System Parameters
| Parameter | Value |
|---|---|
| Molecule | H₂ |
| Bond length | 0.735 Å |
| Basis set | STO-3G |
| Qubits | 4 |
| Exact energy | -1.1373060358 Ha |
| Nuclear repulsion | 0.71996899 Ha |
| Measurement bases | 5 |
6. Files Produced
docs/
└── QCOS_WHITEPAPER_v1.0.md # Full white paper (531 lines)
tests/api-test/benchmarks/
├── scientific_protocol.py # Core VQE (1,588 lines)
├── test_pauli_convention.py # 35 unit tests (680 lines)
├── test_qcos_features.py # Feature validation (550 lines)
└── validate_hamiltonian_equivalence_v2.py # Proof script
results/scientific/
├── hamiltonian_equivalence_v2_*.json # Spectral proof
├── qcos_feature_adaptive_shots_*.json
├── qcos_feature_multi_start_*.json
├── qcos_feature_warm_start_*.json
└── fair_vqe_experiment_*.json
7. Commands to Reproduce
# Run all unit tests
python -m pytest test_pauli_convention.py -v
# Run Hamiltonian equivalence proof
python validate_hamiltonian_equivalence_v2.py
# Run QCOS feature validation (20 runs)
python test_qcos_features.py --runs 20
# Run specific test
python test_qcos_features.py --test adaptive --runs 10
8. Next Steps
- Larger Molecules: Extend to LiH, BeH₂
- Real Hardware: IBM Quantum, IonQ validation
- Adaptive Shots: Variance-based allocation
- Publication: Submit to arXiv/journal
Version: 1.0 Date: January 30, 2026 Status: VALIDATION COMPLETE ✅