Learn · the math curriculum

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The mathematical foundations, as interactive pages: from eigenvalues to covariance to the SPD manifold, and from there to density matrices and the Bloch sphere. Foundations first, geometry second, quantum third.

  1. 01
    Eigenvalues & EigenvectorsThe natural axes of a matrix. Eigenvectors are special directions that don’t rotate under transformation; eigenvalues measure how much they stretch.
    foundational
  2. 02
    Covariance MatricesHow EEG channels relate to each other. Each window of brain activity becomes a covariance matrix capturing correlation structure between electrodes.
    foundational
  3. 03
    SPD MatricesSymmetric Positive Definite matrices form a curved manifold. Every valid covariance matrix lives on this manifold. Explore the boundary where matrices become singular.
    intermediate
  4. 04
    Phase Locking ValueHilbert-transform phase synchrony encoded as quantum rotation angles. The direct bridge between raw EEG and the A-Gate circuit.
    intermediate
  5. 05
    Riemannian GeometryThe SPD manifold is curved, not flat. Geodesics are the shortest paths; tangent spaces let us do linear algebra locally.
    intermediate
  6. 06
    Bures ManifoldQuantum density matrices form their own manifold with the Bures metric. The structural identity between SPD and Bures manifolds explains why quantum circuits can read covariance geometry.
    advanced
  7. 07
    Quantum StatesQubits in superposition on the Bloch sphere. Measurement collapses superposition to definite outcomes with specific probabilities.
    intermediate