We introduce a split-interval representation of operator order in a unital C∗-algebra. The two productsBA and AB are realized as the two sides of a single continuous section on the classical double-arrowinterval. Commutativity is equivalent to descent through the projection to the ordinary interval, and one-halfof the commutator norm is the exact distance to the descent subspace. For finite unitary families these localobstructions weight the adjacent-transposition Cayley graph of Sn. The resulting bottleneck pseudoultrametricadmits a closed inversion-set formula, eliminating a factorial path search. Threshold classes are described byweighted incompatibility graphs, and the marked geometry recovers the complete pairwise commutator-normmatrix. We construct parallel additive Lie-commutator and multiplicative group-commutator enrichments.Exact contextual multiplicative transport is flat, whereas a pair-reduced transport obtained by discardingprefix conjugations can carry nontrivial square and braid-face holonomy. Near the identity, the leadingpossible reduced face defect is fourth order and is governed by commutators of commutators; an exactspin- 12example has nontrivial braid holonomy. For unitary quantum circuits, every reduced adjacent-swappath is simultaneously optimal for all pair-weighted ℓp swap costs, and precedence-constrained thresholdreordering reduces exactly to DAG acyclicity. Reproducible QAOA and Heisenberg/XYZ benchmarks exhibitthe resulting factorial-to-polynomial reduction in classical reordering
We introduce a split-interval representation of operator order in a unital C∗-algebra. The two productsBA and AB are realized as the two sides of a single continuous section on the classical double-arrowinterval. Commutativity is equivalent to descent through the projection to the ordinary interval, and one-halfof the commutator norm is the exact distance to the descent subspace. For finite unitary families these localobstructions weight the adjacent-transposition Cayley graph of Sn. The resulting bottleneck pseudoultrametricadmits a closed inversion-set formula, eliminating a factorial path search. Threshold classes are described byweighted incompatibility graphs, and the marked geometry recovers the complete pairwise commutator-normmatrix. We construct parallel additive Lie-commutator and multiplicative group-commutator enrichments.Exact contextual multiplicative transport is flat, whereas a pair-reduced transport obtained by discardingprefix conjugations can carry nontrivial square and braid-face holonomy. Near the identity, the leadingpossible reduced face defect is fourth order and is governed by commutators of commutators; an exactspin- 12example has nontrivial braid holonomy. For unitary quantum circuits, every reduced adjacent-swappath is simultaneously optimal for all pair-weighted ℓp swap costs, and precedence-constrained thresholdreordering reduces exactly to DAG acyclicity. Reproducible QAOA and Heisenberg/XYZ benchmarks exhibitthe resulting factorial-to-polynomial reduction in classical reordering
We study what happens when an exactly flat contextual operator-order transport is reducededgewise by a unital completely positive (UCP) map. The relevant obstruction is not nonunitarityby itself but failure of multiplicativity. For a UCP map Φ we use the standard multiplicativitydefect∆Φ(X, Y ) = Φ(XY ) − Φ(X)Φ(Y )and prove an exact decomposition of every reduced flat-loop defect into transported multiplicativity defects. The same defect obeys an exact composition law under nested reductions and hasa Stinespring leakage representation. For a uniformly continuous UCP semigroup Φt = etL, themultiplicativity defect satisfies an exact Duhamel formula, and for a bounded GKSL generatorthe corresponding Leibniz defect isΓL(X, Y ) = Xα[V∗α , X][Y, Vα].This yields an exact integral representation of reduced loop holonomy and explicit qubit braid-loopformulas for dephasing, depolarization, and amplitude damping.We then study the inverse problem. Closed-loop finite-channel data need not identify anarbitrary UCP channel, whereas the full infinitesimal Leibniz defect determines a boundedfinite-dimensional generator modulo Hamiltonian derivations. In Md(C) the dissipative quotienthas real dimension (d2 − 1)2, giving the face-count lower bound LCoxd ≥ ⌊d2/2⌋. Using blockextension, centered scalar-one tangent geometry, binder-compatible transversality, a transversenative-tilt argument, and a determinant-one odd-to-even transfer, we prove the matching upperbound. Thus, in the declared bounded finite-dimensional matrix-valued first-order Coxeter-facemeasurement model,LCoxd =d22, d ≥ 3.This sharp count is not a universal lower bound for arbitrary process tomography or arbitraryLindbladian-learning protocols. Conditioning, sample complexity, noise robustness, and finitetime channel reconstruction remain separate questions.
We establish a necessary condition for quantum heuristics to achieve nontrivial computational advantage relative to a classical problem generator A. We prove that any unitary procedure whose dynamics nearly commute with A is spectrally confined: its ability to transfer amplitude across separated spectral sectors is bounded strictly by the commutator defect divided by the spectral gap. This yields a quantitative operator-theoretic obstruction to global state-space exploration. Consequently, repeated application of such procedures (e.g., in deep variational circuits) produces apparent algorithmic activity without generating meaningful information gain. We classify this regime as a "degenerate quantum heuristic". We apply this framework to Variational Quantum Eigensolvers (VQE) and the Quantum Approximate Optimization Algorithm (QAOA), demonstrating that circuit depth cannot overcome transversality failure. The theoretical results are fully closed within trace-ideal (S₂) operator theory. We extend the spectral confinement bound to the entire resolvent geometry, proving that degenerate heuristics preserve the Birman-Schwinger kernel and regularized Fredholm determinants (det₂), thereby mathematically obstructing any computational phase transition or new eigenvalue emergence. Finally, we demonstrate that genuine quantum speedup requires structural noncommutativity, which can be deterministically enforced via an Ω-Σ dissipative control layer. The manuscript includes complete finite-dimensional numerical validation, exhibiting the exact quantitative rigidity between commutator defect, spectral transport, and resolvent deformation.
The analysis of Hamiltonian Variational Ansätze centers on three fundamental questions: which states are exactly reachable from a given initial state, what native control depth suffices for local and global synthesis, and whether a bounded circuit extension can expose a direction of decreasing cost at a suboptimal output. We address these challenges by leveraging the dynamical Lie algebras (DLAs) associated with the circuits and the geometry of the initial state's orbit under the action of the corresponding dynamical Lie group G. We show that an initial state has a closed projective orbit whenever it is a nondegenerate eigenstate of a control Hamiltonian. Combined with asymptotic QAOA convergence, this guarantees exact, finite depth ground state preparation for architectures employing diagonal problem Hamiltonians and irreducible stoquastic mixers. We bound the number of native control exponentials required for local unitary synthesis and state preparation using DLA commutator filtration growth alongside reachable orbit dimensions. Complementing these local results, we construct global frameworks on invariant sectors equipped with full unitary or special unitary dynamical Lie algebras. These constructions yield explicit applications to Pauli-generated systems and constrained maximum independent set multi-angle QAOA circuits; specifically, we show that their native control costs are directly determined by commutator filtration depth and independent-set toggle-group geometry, respectively. Finally, if the dynamical Lie group is compact, the observable HP generates an admissible evolution (iHP∈g, where g is the DLA), and the projective orbit of the initial state contains a ground state ray of HP, then a bounded depth, identity-initialized circuit extension exhibits either first-order descent or negative curvature at every suboptimal circuit output.
The quantum switch is known to probe order-sensitive operator structure, including commutators, incompatibility measures, and unitary invariants \citep{Chi13,Ore12,Pro14,Gao22,Aza25}. Related indefinite-order protocols can read out loop-like geometric phases \citep{Zha20,Yin23,Bar24}. This paper develops the gauge-holonomy case. Let Uγ1 and Uγ2 be path-ordered holonomies of two loops based at a common point x0 and represented unitarily on a target Hilbert space. For a standard two-branch quantum switch, with one branch applying U then V and the other applying V then U, the lower off-diagonal control coherence is\[\chi=\Tr\!\left(K(U,V)\rho\right),\qquadK(U,V)=U^\dagger V^\dagger U V.\]Thus the switch reads out a based holonomy group commutator. Scalar Abelian holonomies and, more generally, commuting holonomies are order-invisible in the bare switch. Abelian parameter sensitivity can occur only through relative encoding or reference structure. Non-Abelian holonomies are directly visible through control-qubit quadratures. The finite-loop statement is exact and gauge covariant: under a common base-point transformation, K↦gKg−1, while traces and covariantly prepared target expectations are gauge invariant. The local small-loop reduction is separate and perturbative. For small based loops with logarithms $X=\Log U$ and $Y=\Log V$,logK(U,V)=[X,Y]−21[X+Y,[X,Y]]+terms of degree ≥4.For loops contained in a coordinate patch, X begins with the area contraction of the curvature, followed by loop-shape-dependent covariant-derivative and path-ordering corrections. The leading local contribution to logK is therefore a curvature-commutator term, but only in this controlled small-loop regime. A constructive gauge-loop realization, an SU(2) signal model, a lattice-gauge analogue, and an experimental readout model are given. The result is an operator-level and gauge-theoretic proposal for using switch interferometry to read out non-Abelian holonomy order interference.
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 S(ε) from trace data of short group-commutator words and use it to drive an explicit native-operation promotion protocol. The bracket-norm interpretation of S(ε) is asymptotic with a general O(ε) remainder; for inversion-symmetric primitive families all odd-order corrections vanish, giving O(ε2) 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 ΔPθ = C(θ)HE(χ, γ), with C(θ) the concurrence and HE = sin2(2γ) sin4(χ/2) 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 −σx on span{|20⟩, |02⟩}; 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. This record contains the abstract only. It is not the full text of the article. The complete peer-reviewed article is published open access in Discover Quantum Science and is freely available from the publisher at https://doi.org/10.1007/s44464-026-00037-5 — including all figures, tables, appendices and the supplementary code and data. Cite as: Kazadi J. Holonomy-first native operations geometric quantum framework and entanglement-sensitive multi-particle holonomic gates. Discover Quantum Science. 2026;2:35. https://doi.org/10.1007/s44464-026-00037-5
We develop a systematic framework for quantum mechanics on a finite graph-resolved class of second-countable, T1, non-Hausdorff one-manifolds. The central problem is that local differential expressions do not by themselves determine the quantum theory: non-Hausdorff incidence, bundle transport, smooth extension, and analytic completion can impose additional global constraints on the operator domain. We formulate these constraints using transported jets, Whitney realisability, Sobolev traces, and formal return holonomy, obtaining exact closure results for finite smooth graph resolutions.
At first Sobolev order, scalar dynamics reduce to a weighted quantum graph whose edge weights arise from a marked resolving presentation, leading naturally to weighted Kirchhoff Hamiltonians. Multiple-origin lines and circles, branching junctions, finite trees, and split-and-rejoin geometries then yield explicit deficiency indices, reflection and transmission laws, interference conditions, compact dark states, and embedded cavity modes.
For finite-rank Hermitian bundles, unitary gluing selects a transmitting fixed subspace, so scattering becomes projection onto the invariant sector of the subgroup generated by the gluing matrices. Compact-group representations therefore turn non-Hausdorff incidence into an invariant-sector selection mechanism; connected compact semisimple groups require at most two relative gluing matrices for full invariant completion, with explicit SU(3) examples.
The resulting theory separates topological non-Hausdorff data from the analytic and representation-theoretic structures that remain detectable by quantum dynamics.