Math Concepts | QDNU

Understanding the A-Gate architecture requires familiarity with several mathematical concepts from linear algebra, differential geometry, and quantum computing. These interactive visualizations build intuition for each concept, progressing from foundational to advanced topics.

Foundational Concepts

Covariance Matrices Foundational

How EEG channels relate to each other. Each 1-second window of brain activity becomes a covariance matrix capturing the correlation structure between electrodes.

Interactive EEG Context 4 Scenes

Eigenvalues & Eigenvectors Foundational

The natural axes of a matrix. Eigenvectors are special directions that don't rotate under transformation; eigenvalues measure how much they stretch.

Interactive Decomposition 4 Scenes

SPD Matrices Intermediate

Symmetric Positive Definite matrices form a curved manifold. Every valid covariance matrix lives on this manifold. Explore the boundary where matrices become singular.

Interactive Slider Ellipse Viz Quadratic Form

Geometric Concepts

Riemannian Geometry Intermediate

The SPD manifold is curved, not flat. Geodesics are the shortest paths; tangent spaces let us do linear algebra locally. Understanding this curvature is key to measuring brain state distances.

Three.js Geodesics 5 Scenes

Phase Locking Value Intermediate Advanced

Hilbert-transform phase synchrony encoded as quantum rotation angles. How PLV quantifies cross-electrode synchrony — the direct bridge between raw EEG and the A-Gate circuit.

Hilbert Transform Phase Synchrony 4 Scenes

Quantum Concepts

Quantum States Intermediate

Qubits exist in superposition of |0> and |1>. The Bloch sphere visualizes single-qubit states. Measurement collapses superposition to definite outcomes with specific probabilities.

Three.js Bloch Sphere 5 Scenes

Bures Manifold Advanced

Quantum density matrices form their own manifold with the Bures metric. The deep connection between SPD and Bures manifolds explains why quantum circuits can detect EEG polarity.

Density Matrices Fidelity 4 Scenes

A-Gate Architecture Advanced

The complete 17-qubit circuit: how Phase Locking Values become rotation angles, how entangling gates mix information, and why opposite outcomes reveal hidden polarity.

Circuit Diagram Encoding 5 Scenes

Geometric Invariants Advanced

The invariant hierarchy on the SPD manifold: spatial covariance polarity (confirmed), cross-frequency coupling (predicted), and manifold trajectory geometry (predicted). Testable predictions with specific falsification criteria.

Hierarchy Cross-Frequency 4 Scenes

← Back to qdnu.ai Launch Visualization →