Distributed Verification of Lie Closure (DVLC): Gauge-invariant certification from holonomy class functions

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.