This work introduces Distributed Verification of Lie Closure (DVLC), a gauge-invariant certification protocol for testing whether a set of experimentally accessible control primitives generates a desired target Lie algebra. DVLC is designed for settings—such as holonomic and geometric quantum control—where only conjugacy-invariant data (e.g. traces, characters, Wilson-loop observables) are naturally available, and where full Hamiltonian reconstruction or explicit commutator closure tests are impractical. Rather than reconstructing generators or computing nested commutators, DVLC compares experimentally estimated class functions of short, near-identity word holonomies against universal Baker–Campbell–Hausdorff (BCH) predictions for a chosen compact target group (e.g. SU(d)). The protocol operates directly on the adjoint quotient (conjugacy classes), avoiding gauge fixing by construction, and parallelises naturally across words and invariants. Under explicit and physically transparent assumptions—local separation of conjugacy classes by finitely many class functions, uniform BCH truncation on a finite word suite, and a finite Lie presentation of the target algebra by low-degree relators—a soundness theorem is proven: agreement of invariant data implies that the system generators satisfy the same defining relators as the target, inducing a Lie-algebra homomorphism into the system (and an embedding for simple targets). Quantitative robustness results, including explicit stability for SU(d) trace invariants, and finite-sample, distribution-agnostic guarantees suitable for distributed experiments are provided. Minimal SU(2) and SU(3) examples demonstrate how DVLC detects noncommutativity and higher-bracket structure using only a small number of scalar, gauge-invariant observables—without tomography, commutator computation, or basis-dependent rank conditions. This preprint is intended as a self-contained theoretical contribution to quantum control, holonomic quantum computation, and Lie-algebraic certification, and the author welcomes feedback and discussion.
This work introduces Distributed Verification of Lie Closure (DVLC), a gauge-invariant certification protocol for testing whether a set of experimentally accessible control primitives generates a desired target Lie algebra. DVLC is designed for settings—such as holonomic and geometric quantum control—where only conjugacy-invariant data (e.g. traces, characters, Wilson-loop observables) are naturally available, and where full Hamiltonian reconstruction or explicit commutator closure tests are impractical. Rather than reconstructing generators or computing nested commutators, DVLC compares experimentally estimated class functions of short, near-identity word holonomies against universal Baker–Campbell–Hausdorff (BCH) predictions for a chosen compact target group (e.g. SU(d)). The protocol operates directly on the adjoint quotient (conjugacy classes), avoiding gauge fixing by construction, and parallelises naturally across words and invariants. Under explicit and physically transparent assumptions—local separation of conjugacy classes by finitely many class functions, uniform BCH truncation on a finite word suite, and a finite Lie presentation of the target algebra by low-degree relators—a soundness theorem is proven: agreement of invariant data implies that the system generators satisfy the same defining relators as the target, inducing a Lie-algebra homomorphism into the system (and an embedding for simple targets). Quantitative robustness results, including explicit stability for SU(d) trace invariants, and finite-sample, distribution-agnostic guarantees suitable for distributed experiments are provided. Minimal SU(2) and SU(3) examples demonstrate how DVLC detects noncommutativity and higher-bracket structure using only a small number of scalar, gauge-invariant observables—without tomography, commutator computation, or basis-dependent rank conditions. This preprint is intended as a self-contained theoretical contribution to quantum control, holonomic quantum computation, and Lie-algebraic certification, and the author welcomes feedback and discussion.
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.
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.
We present a self-contained, auditable framework for reconstructing discrete gauge con nections from loop-based records (e.g. interferometric phase/visibility readouts) on a finite 2- complex. The core contribution is a Minimal Verification Suite (MVS) that separates three logically distinct questions: (i) curvature consistency via plaquettes, (ii) non-commutativity via commutator loops, and (iii) constraint compatibility via a discrete Gauss closure defect. We prove that a commutator suite passing on a generating set implies Abelian reducibility of the reachable subgroup, and we specialize this statement to SU(2), showing commuting controls lie in a common maximal torus (a single rotation axis). A central technical result is an exact closed-form identity for the SU(2) commutator Wilson scalar: Wcomm = 1 2Tr(UV U −1V −1 ) = 1 − 2 sin2 (α/2) sin2 (β/2) ∥n × m∥ 2 , where U = exp(iα n · σ/2) and V = exp(iβ m · σ/2). This yields a hard, record-only axis misalignment bound under a measurable non-degeneracy gate. We then harden MVS sound ness into explicit, certificate-friendly inequalities: loop-fit errors propagate to chord-edge errors without path amplification in spanning-tree gauge, Projects can be compared by a constructive gauge-alignment algorithm, and Gauss-closure defects admit a calibrated envelope law decom posing statistical, calibration, leakage, and non-Abelian contributions. Finally, we introduce tiered certificates (MVS-A Abelian, MVS-N Non-Abelian, MVS-B Boundary-limited) and pro vide a fully worked 2 × 2 plaquette patch example with replayable YAML templates.
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.
Determining the unitary dynamics accessible from finite Hamiltonian resources is a central problem in Hamiltonian engineering and quantum control. Dynamical Lie algebras (DLAs) connect available control Hamiltonians with the reachable dynamics, but their use as a design tool for modifying Hamiltonian generator sets remains less developed. In this work, we develop a finite-dimensional DLA framework for three generator-set operations: composition, invariance, and reduction. For composition, we construct direct sums of component DLAs using spectral projectors on an auxiliary register. For invariance, we analyze when modifications of Pauli-string generating sets preserve the generated Lie algebra, and introduce algebraic diagnostics for added generators. For reduction, we consider compact reductive DLAs and show how projection onto selected simple ideals gives reduced generating sets whose Lie closures are the corresponding ideal sums. We illustrate these results with finite-dimensional examples and numerical checks, including direct-sum dimension addition, central-spin invariance diagnostics, and DLA-based ansatz reduction for block-local Hamiltonians. The results show how DLA structure can be used to diagnose controllability and guide Hamiltonian generator design under explicit algebraic assumptions.