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
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
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 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 preprint presents a unified holonomy-first, invariant-only simulator architecture for loop-based quantum operations, integrating two complementary components:(i) an operational framework for discovering, diagnosing, and compiling native holonomic operations from closed-loop executions, and(ii) a minimal semantic language that characterizes what such loop-based operations mean when only conjugacy-invariant data are accessible. The simulator treats holonomy words—compositions of calibrated loop primitives—as the fundamental operational objects, rather than Hamiltonians, generators, or gauge-fixed operators. All semantics are defined at the level of conjugacy classes, represented operationally by invariant fingerprints such as trace moments or overlap-derived scalars. This invariant-first stance enables robust workflows that avoid full operator reconstruction, tomography, or fixed-basis assumptions. On this substrate, the simulator introduces a geometric compilation pipeline that: Extracts interaction structure from short loop compositions using commutator and nested-commutator diagnostics; Organizes noncommutativity and curvature information into a gauge-invariant interaction spectrum; Promotes reproducible, scale-consistent loop words into reusable native operations. A central contribution of the unified framework is a theorem-level semantic obstruction: the default single-system semantic tier is generically incomplete in the presence of entanglement-sensitive loop effects. This is demonstrated by introducing entanglement holonomy, a word-level invariant accessible via joint return amplitudes of entangled probes yet invisible to any reconstruction based solely on single-system class functions or mode-preserving data. As a consequence, the simulator treats the effective representation on which holonomies act as provisional rather than fixed. When invariant closure and scaling tests fail at a given tier, the simulator is forced to lift representation (e.g., from single-particle to multi-particle or encoded subspaces) as an operational necessity rather than a modeling choice. An explicit minimal example of an irreducible two-particle holonomic gate is provided within a quadratic-mode backend. The simulator does not attempt to unify classical and quantum dynamics or to provide a theory of quantum gravity. Instead, it supplies a shared holonomy-based semantic layer in which classical geometric transport and genuinely quantum effects (including entanglement) can be jointly represented and operationally distinguished using the same invariant diagnostics. This unified manuscript synthesizes and extends two previously developed components—an operational framework (GQF, DOI: 10.5281/zenodo.18303288) and a semantic foundation with an obstruction result (MHQL, DOI: 10.5281/zenodo.18332083)—to expose a structural point not apparent when they are treated separately: invariant-first compilation and representation lifting are linked by a principled semantic incompleteness theorem. The result is a rare simulator architecture that is holonomy-driven, gauge-invariant by construction, and representation-adaptive when compelled by invariant evidence.
MHQL (Minimal Holonomic Quantum Language) provides the semantic foundation for holonomy-first, invariant-only workflows such as the Geometric Quantum Framework (GQF). DOI: 10.5281/ZENODO.18303288 Where GQF introduces an operational methodology for discovering, diagnosing, and promoting native holonomic operations from closed-loop executions, MHQL addresses a logically prior question: What does a loop-based operation mean when only conjugacy-invariant data are accessible, and when is a given semantic tier complete? MHQL formalizes a minimal language of holonomy words, built from calibrated loop primitives and composed using products, inverses, and commutators. The semantics of this language are defined entirely at the level of conjugacy classes, represented operationally by invariant fingerprints (trace moments, overlap-type scalars), without gauge fixing, basis choices, or full operator reconstruction. The contribution of MHQL is not a new geometric phase, a new holonomic gate, or a new control model. Instead, it provides a semantic framework and obstruction result that clarifies the limits of invariant-only reasoning: It defines a minimal, invariant-only semantic layer for loop-based operations; It formalizes small-loop ordering and invariant consistency checks (echoes, palindromes, commutators); It proves that the default single-system semantic tier is incomplete in the presence of entanglement-sensitive loop effects; It introduces entanglement holonomy as a word-level invariant that is operationally accessible yet invisible to any reconstruction based solely on single-system class functions or mode-preserving data. This semantic no-go result directly complements GQF’s operational stance. In GQF, the effective representation on which holonomies act is intentionally left flexible and is promoted only when invariant closure and scaling tests fail. MHQL explains why such promotion must sometimes occur, while GQF provides the diagnostic and operational machinery for detecting this situation and acting on it. Together, MHQL and GQF form a coherent holonomy-first stack: MHQL defines what is meaningful at the invariant level and establishes the semantic limits of single-system descriptions. GQF shows how to discover, certify, and reuse holonomic operations using exactly those invariant-only tools, and how to lift representations when forced by the data. MHQL is therefore not an alternative to GQF, but its semantic and logical foundation. It clarifies when invariant-first workflows are sufficient, when they are not, and why representation lifting in GQF is a semantic necessity rather than a modeling choice.
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.
We define a basis-independent two-particle return-amplitude residue H_E = sin²(2γ) sin⁴(χ/2), constructed from a single-particle return angle χ ∈ [0, π] and a local dyad rotation γ. The residue is bounded in [0, 1], scales quartically for small χ, and isolates an off-diagonal cross term that appears in the symmetric entangled return amplitude but not in the corresponding product-state amplitude. It is invariant under common unitary changes of basis at the basepoint and depends only on operational overlap data, not on the mechanism that produces the loop. A minimal number-conserving two-boson hopping Hamiltonian is then shown to carry a nontrivial encoded holonomic gate. The encoded subspace span{|20⟩, |02⟩} satisfies the scalar-compression condition with vanishing compressed Hamiltonian, and at T = π/(2J) the cyclic return map is U_code = −σ_x, purely geometric in the projected-evolution sense. The associated one-particle obstruction is calibrated using higher-rank numerical ranges. For a Hermitian one-body Hamiltonian κ on m modes, the rank-2 scalar-compression condition is governed by Λ_2(κ) = [λ_2, λ_{m−1}]. The obstruction holds at m = 2 unless κ is scalar, collapses to a single value at m = 3, and becomes a generic interval at m ≥ 4. The two-boson construction is therefore a minimal mode-resource separation, not an absolute single-particle no-go theorem. The paper also gives an interferometric estimation protocol for H_E, the resulting shot-count scaling, and a comparison with Lusk's spacetime-specific entanglement holonomy in Kerr geometry.
UHQL (Universal Holonomic Quantum Language) is a hardware-independent, gauge-covariant framework for holonomic quantum computation. It reformulates geometric quantum control as a symbolic language whose elementary operations are closed-loop holonomies acting on a finite-dimensional logical subspace. The core of UHQL is an alphabet of 22 primitive holonomies, each realized as a calibrated closed control loop on a ten-dimensional logical Hilbert space. Programs are expressed as words generated by concatenation, inversion, and commutators of these primitives. The physical semantics of a program are defined only up to conjugacy, ensuring gauge covariance and portability across distinct hardware platforms. A central theoretical result is a Lie–closure universality theorem: if the infinitesimal generators associated with the primitive holonomies span the Lie algebra su(10), then finite UHQL words are dense in SU(10), providing universal geometric control of the logical subspace. This links universality directly to the curvature structure of the Wilczek–Zee connection and to classical controllability results in geometric control theory. The manuscript also introduces DVLC (Distributed Verification of Lie Closure), a gauge-invariant and experimentally feasible protocol for certifying universality using class-function observables (such as traces and Wilson-loop invariants), without explicit reconstruction of the underlying Lie algebra or Hamiltonian. UHQL separates logical structure from device-specific implementation: the same symbolic program can be compiled into calibrated control loops on superconducting, trapped-ion, photonic, or neutral-atom platforms. Robustness follows naturally from the geometric nature of holonomies, including reparametrization invariance, thin-ribbon cancellation of geometric errors, and echo-based suppression of dynamical phases. This deposit corresponds to a preprint / working manuscript. Further refinements and extensions are planned for a future journal submission.