Holonomy-first native operations geometric quantum framework and entanglement-sensitive multi-particle holonomic gates

Abstract

Holonomy-first quantum frameworks treat closed-loop transformations—holonomies—as the primitive operations from which native gates are mined, rather than as derived quantities. We develop this viewpoint in two parts. First, we construct a gauge-invariant commutator-strength matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S(\epsilon )$$\end{document} from trace data of short group-commutator words and use it to drive an explicit native-operation promotion protocol. The bracket-norm interpretation of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S(\epsilon )$$\end{document} is asymptotic with a general \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\epsilon )$$\end{document} remainder; for inversion-symmetric primitive families all odd-order corrections vanish, giving \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\epsilon ^2)$$\end{document} accuracy, and a counterexample shows the symmetry cannot be dropped. The promotion criteria are operationalized as declared statistical tests—sampling-cost laws for interferometric and readout-floor regimes, a bias-corrected scale-equivalence rule, and multiplicity control—and a deterministic simulation campaign reports decision replication, false-promotion control, inverse-defect detection, and execution cost under finite-shot readout, calibration error, imperfect inverses, and visibility loss. Second, for two-particle preparations of Schmidt rank two, the common-basepoint return-amplitude residue factorizes exactly as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta P_\theta =C(\theta )H_E(\chi ,\gamma )$$\end{document}, with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C(\theta )$$\end{document} the concurrence and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_E=\sin ^2(2\gamma )\sin ^4(\chi /2)$$\end{document} a calibrated loop factor, determining negativity, entanglement entropy, product fidelity, and geometric entanglement within the family. A two-boson construction realizes the encoded geometric gate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\sigma _x$$\end{document} on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textrm{span}}\{\left| 20\right\rangle ,\left| 02\right\rangle \}$$\end{document}; under the same two-mode, same-dimension projected-scalar condition, no corresponding one-particle holonomic cycle exists unless the one-body Hamiltonian is scalar. The mathematical ingredients are established; the contribution is the invariant-readout protocol, its quantified cost and uncertainty analysis, and precise statements of what can and cannot be inferred without full operator reconstruction.