Quantum Geometric Computation (QGC)

An invariant-first framework for quantum and classical computation

QGC computes quantum observables through geometric invariants—moments, Gram structures, and curvature measures—rather than exponential state-vector enumeration. The framework provides polynomial-time algorithms for problems traditionally requiring exponential resources.

Core Idea

Instead of tracking 2ⁿ amplitudes, QGC extracts answers from low-order invariants of the Hamiltonian or density matrix:

Traditional: |ψ⟩ → evolve → measure → statistics
QGC:         H → moments → geometry → observables

The key insight is that many quantum observables depend only on global geometric properties (traces, cycles, curvature) that can be computed efficiently.

Framework Components

Universal Composition Engine (UCE)

Computes power-sum moments Tr(ρᵏ) via Cayley-Hamilton recurrence relations, avoiding explicit matrix powers.

Universal Laws (UL-2 → UL-7)

Exact algebraic identities relating moments to physical observables:

• UL-2: Purity from pair correlations
• UL-3: Triple-phase / Bargmann invariant
• UL-4: Quartet correlations and 4-cycles
• UL-5: SU(2) certainty law (step-exact Born rule)
• UL-6/7: Higher-order motif expansions

All validated to machine precision (10⁻¹² – 10⁻¹⁵).

The κ Invariant

A geometric complexity parameter measuring "how much energy is in correlations":

κ = ‖offdiag(M)‖ / ‖M‖

Critical threshold at κ ≈ 0.85 marks transitions between:

• Efficient geometric computation (κ < 0.85)
• Required escalation to higher-order methods (κ ≥ 0.85)

This threshold has been validated across multiple domains.

Repository Structure

├── Core-Files/
│   └── universal_composition_engine.py   # UCE implementation
├── Examples/
│   ├── Grover-Benchmark/                 # Grover's algorithm via invariants
│   ├── Hubbard-Model/                    # 2D Hubbard solver (~0.5% error)
│   └── QAOA-Demo/                        # QAOA MaxCut prediction
├── Proofs-Validations/
│   ├── proof_01_moment_cycle.py          # Tr(ρᵏ) = Tr(Gᵏ)/Nᵏ identity
│   ├── proof_02_purity_bridge.py         # UL-2 validation
│   └── proof_03_kappa_physics.py         # κ threshold behavior
├── GLOSSARY.md                           # Term definitions
├── QGC_White_Paper.md                    # Full technical description
└── README.md

Getting Started

# Clone the repository
git clone https://github.com/tonyboutwell/Quantum-Geometric-Computing.git
cd Quantum-Geometric-Computing

# Try the Hubbard solver
cd Examples/Hubbard-Model
python qgc_hubbard_model_explorer.py bench

# Run proof validations
cd ../../Proofs-Validations
python proof_01_moment_cycle.py
python proof_02_purity_bridge.py

Requirements

• Python 3.8+
• NumPy
• SciPy

Current Status

QGC is an active research project.

References

• Simons Collaboration, Phys. Rev. X 5, 041041 (2015)
• Qin, Shi, Zhang, Phys. Rev. B 94, 085103 (2016)
• Lieb & Wu, Phys. Rev. Lett. 20, 1445 (1968)

Contact

Tony Boutwell
Director of AI and Creative Technologies
Meridian Community College

[email protected]

License

Research use permitted with attribution.