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Implement `clifford_to_transvections_minimal`, decomposing a Clifford into a minimal-length ordered product of Clifford transvections (pi/4 Pauli exponents) reproducing its symplectic action with r or r+1 factors (r = residue rank). The core is an exact, complete congruence-triangularization of the residue form E via memoized backtracking: the paper's forward-greedy pivot search is incomplete at rank >= 5 (it can dead-end on a bad non-isotropic pivot even when a triangularization exists), and its line-411 claim that every non-hyperbolic form is triangularizable is incorrect. The r+1 branch appends a fix vector that restores triangularizability at the same rank. Subspaces proven unsolvable are memoized by canonical RREF key to keep the search tractable. Add Python binding `CliffordUnitary.to_transvections_minimal`, .pyi stub, and Rust + hypothesis tests (symplectic-action roundtrip, minimality bound r/r+1, minimal <= greedy length). Track the regression seed for the previously-failing n=4 case. Co-authored-by: Copilot <223556219+Copilot@users.noreply.github.com> Co-authored-by: Claude Opus 4.8 <noreply@anthropic.com>
Document two issues found while implementing clifford_to_transvections_minimal: 1. The paper's claim that a non-hyperbolic symplectic map's residue matrix is always congruence-triangularizable (hence decomposes into exactly r transvections) is false over F2. The clean r/(r+1) dichotomy is Dieudonne's theorem (O'Meara, Symplectic Groups, Thm 2.1.11), stated only for F != F2; O'Meara's Sec 2.3 explicitly notes it fails over F2. We give a machine-checked 2-qubit counterexample (residue rank 3, minimal length 4) and a census showing 210 non-hyperbolic maps in Sp(4,2) require r+1. 2. Even when a triangularization exists, the forward-greedy pivot search is incomplete for r >= 5; fixed by complete memoized backtracking. States the adopted criterion (min = r iff the residue core is congruence-lower-triangularizable, else r+1) with honest caveats about exact minimality for large m. All references verified against the sources. Co-authored-by: Copilot <223556219+Copilot@users.noreply.github.com> Co-authored-by: Claude Opus 4.8 <noreply@anthropic.com>
Ensure the Clifford -> transvection decomposition (arXiv:2102.11380) is referenced across docs, READMEs, and examples: - Add a runnable Jupyter example (bindings/python/examples/clifford-transvection-decomposition.ipynb) showing greedy vs minimal decomposition, rebuild + symplectic-action verification, the r / r+1 factor count, and the non-hyperbolic r+1 case from the correction note. - Reference the decomposition in paulimer/README.md (Clifford features + documentation list) and in the Python bindings README (feature bullet + quick-start snippet). - Make this branch self-contained: rephrase the intra-doc links to clifford_to_pauli_exponents / to_pauli_exponents (which live on a separate branch) into plain text, so cargo doc builds standalone. Co-authored-by: Copilot <223556219+Copilot@users.noreply.github.com> Co-authored-by: Claude Opus 4.8 <noreply@anthropic.com>
Generate residue-space candidates lazily without fixed-width masks, preventing eager exponential allocation and rank-64 shift overflow while preserving exhaustive search order. Document the exact search's remaining exponential time and memoization costs. Co-authored-by: Copilot <223556219+Copilot@users.noreply.github.com> Co-authored-by: GPT-5.6 Sol <noreply@openai.com>
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Superseded by microsoft#119, targeting the upstream repository. |
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Summary
rorr + 1factors.Reference
Builds on arXiv:2102.11380, while documenting and correcting its minimality criterion: the minimum is
rexactly when the residue core is congruence-triangularizable, andr + 1otherwise. The included note gives a machine-checked two-qubit counterexample to the paper’s broader claim.Stack
Stack B, PR 2 of 2. Base:
masilv/clifford-to-pauli-exp-unphased. Review and merge after Stack B PR 1; this stack is independent of Stack A.