Add minimum-length Clifford transvection decomposition - #119
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Marcus P S (marcusps) wants to merge 18 commits into
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Implement a linear-size decomposition of a Clifford into Clifford transvections (pi/4 Pauli exponents) reproducing its symplectic (conjugation) action, plus the centralizer generators (the paper's headline application), in the paulimer crate. - paulimer::clifford::clifford_to_transvections: greedy O'Meara-style reduction producing O(n) factors that reproduce the symplectic action (signs and global phase are not tracked; contrast the sign-exact, O(n^2) clifford_to_pauli_exponents). Not guaranteed strictly minimal; exact r/r+1 minimality via congruence triangulation is a follow-up. - paulimer::clifford::clifford_centralizer: generators of Paulis fixed up to sign under conjugation (kernel of the residue map over GF(2), via binar). - Python bindings CliffordUnitary.to_transvections() / .centralizer() with .pyi stubs. - Extensive Rust (unit + proptest) and Python (hypothesis) tests: symplectic round trip, linear factor bound, no-shorter-than-minimum, and centralizer conjugation-fixedness. Co-authored-by: Copilot <223556219+Copilot@users.noreply.github.com> Co-authored-by: Claude Opus 4.8 <noreply@anthropic.com>
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>
Marcus P S (marcusps)
marked this pull request as ready for review
July 17, 2026 04:35
Co-authored-by: Copilot <223556219+Copilot@users.noreply.github.com>
…ford-to-transvections-minimal
Co-authored-by: Copilot <223556219+Copilot@users.noreply.github.com>
Apply the conventions established while addressing @jmbr's review on the phased-outcome / decomposition stack (PRs microsoft#115/microsoft#116) to the transvection decomposition, anticipating the same feedback here: - Replace the seed + internal-RNG proptests (`seed in any::<u64>()` feeding `CliffordUnitary::random`) with a `Gate`-sequence `Strategy`, mirroring phased_clifford_dense.rs and measure_with_hint_sign_test.rs, so a failing input shrinks to a minimal gate sequence instead of an opaque seed. The deterministic sweep (which needs no shrinking) keeps using a seeded RNG. - Derive `qubit_count` inside `next_transvection` from `working.num_qubits()` instead of threading it as a parameter, matching the earlier "derive qubit count" tidy on the Gaussian-elimination decomposition. Co-authored-by: Copilot <223556219+Copilot@users.noreply.github.com> Copilot-Session: 16f923fe-a72a-4430-97b0-b7677f48dfb7
…inimal Update the minimal-decomposition rung with the latest from the lower rung clifford-to-pauli-exp-unphased (79c66e3): the Gate-strategy proptests and the param-less next_transvection tidy, applying the @jmbr review conventions from PRs microsoft#115/microsoft#116/microsoft#117 to this branch too. Also align the branch's own minimal-decomposition tests with those conventions: convert the remaining seed + internal-RNG proptests (minimal_reproduces_symplectic_action, minimal_is_r_or_r_plus_one_and_at_most_greedy) to the shared Gate strategy so they shrink to a minimal gate sequence, and drop the committed transvection_test.proptest-regressions seed file. The deterministic seeded sweeps (which need no shrinking) keep using random_clifford. Co-authored-by: Copilot <223556219+Copilot@users.noreply.github.com> Copilot-Session: 16f923fe-a72a-4430-97b0-b7677f48dfb7
Co-authored-by: Copilot <223556219+Copilot@users.noreply.github.com>
Co-authored-by: Copilot <223556219+Copilot@users.noreply.github.com>
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Summary
rorr + 1factors.Correctness and published reference
Builds on Decomposition of Clifford Gates (arXiv:2102.11380). Theorem 3’s existence conclusion is correct, but its construction incorrectly assumes every non-hyperbolic map has a congruence-triangularizable residue core and therefore a length-
rdecomposition.This PR implements the exact criterion: the minimum is
rprecisely when the full nonsymmetric residue core is congruence triangularizable, andr + 1otherwise. The correction note identifies Callan’s class-A exceptions and retains a deterministic two-qubit regression with residue rank three and minimum length four.The corrected proof was checked independently with an interactive theorem prover. The check establishes the residue-rank lower bound, exact length-
rcriterion, universalr + 1upper bound, strict minimality, and the residue-fix theorem used by the fallback.Review series
This is the second of two dependent PRs and is stacked on #118. Review and merge #118 first. Until it merges and this branch is rebased, GitHub shows its commits cumulatively; this PR’s new review scope is the strict-minimum algorithm, correction note, tests, and examples.