Mathematics · proofs · software

Quantum and Circuits · Visual guides

Explore the 50 visual pages

Choose either complete 25-page sequence. The linked reading pages retain the original drawings, selectable equations, source links, and recorded limits.

Quantum arithmetic

Follow selectors, Fourier coordinates, amplitudes, and measurement through the original worked diagrams.

  1. One arithmetic selector, three realizations
  2. Coprimality selects primitive roots
  3. The selected phases add and cancel
  4. Interference yields an integer signal
  5. Why the phase sum is an integer
  6. The spectrum recovers the selector
  7. A diagonal selector becomes a rational filter
  8. Give the projector something to remove
  9. Account for the entire input
  10. Projection and unitary change are different operations
  11. All twelve modes belong to six order classes
  12. Six exact filters in one ambient space
  13. A packet's capacity is not its occupied energy
  14. The arithmetic signal is a coherent amplitude vector
  15. Amplitudes and probabilities tell different stories
  16. Identical histograms can hide orthogonal states
  17. A basis change makes the hidden distinction observable
  18. Move continuously between the two states
  19. A real three-dimensional view of one amplitude
  20. Rearrange twelve samples into three exact lanes
  21. Phase factors connect the lane spectra
  22. The construction extends beyond this picture
  23. State exactly what each passage preserves
  24. Exact data, floating rendering, and formal coverage
  25. What the construction establishes

Circuit obstructions

Explore legal circuit models, restored boundaries, cut comparisons, and the limits recorded with each example.

  1. The model: boundary restoration and Boolean degree
  2. An arbitrary cut compares two reversible computations
  3. Why completeness is pointwise, and why no decoder is needed
  4. A dirty scratch wire can be fully restored
  5. The output direction can move away from the target
  6. Degree bounds must retain shared register products
  7. The invariant handles duplicates, NOT, and CNOT
  8. One safe packet need not cover every accepting point
  9. Balanced cuts turn coverage into a Toffoli bound
  10. An affine shift reveals a harder accepting point
  11. A degree-one dual certificate for two-input AND
  12. The finite evaluation-span dual in full
  13. Every function through three inputs has a matching oracle
  14. Projection is defined in the original coordinates
  15. Exact nonlinear projection costs versus actual input length
  16. At m=10, the total-gate accounting points the other way
  17. Quadratic projection can destroy safety
  18. Witness indicators are orthogonal idempotents
  19. Faithful linear multiplicative summaries need 2^m dimensions
  20. Distinct witness monomials prevent formal cancellation
  21. A short expression inside a trace is not a computed trace
  22. Disjoint-variable factorization uses each predicate's own block
  23. Overlap breaks the attempted context-free product rule
  24. Randomized detection is conditional on efficient evaluation
  25. Reproduce the figures and inspect their exact evidence

For a shorter suggested route, use the visual introduction. For definitions and proofs, return to the full volume contents.