Linear and non-linear relational analyses for Quantum Program Optimization

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Recasts phase folding as an affine relational analysis and implements nonlinear extensions as affine varieties / polynomial ideals with Gröbner-basis operators, plus a symbolic path integral extractor that produces exact transition ideals for straightline blocks. Prototype implemented in Fineman for open Chasm 3 shows loop invariants and non-trivial optimizations that previous tools missed, most improvements needing degree‑2 polynomial relations (degree unbounded supported but rarer), and artifacts are available and reusable.

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" chapters (13)

1. 0:00Introduction
2. 2:11Quantum data flow
3. 3:57Circuit optimization methods
4. 6:27Semantic challenges in programs
5. 8:09Classical semantics of quantum circuits
6. 10:39Affine relational analysis
7. 13:27Nonlinear reasoning with affine varieties
8. 15:30Handling interference with path integrals
9. 16:53Implementation in Fineman optimizer
10. 18:17Takeaways and future directions
11. 18:41Q&A: Polynomial degree
12. 19:48Q&A: Abstraction ordering
13. 20:29Q&A: Loop invariant inference