Pullback of canonical-graph applications to original instances

Determine to what extent the applications enabled by perfect canonical anticommutation graphs—including entanglement detection, shadow-tomography complexity analysis, uncertainty relations, ground-state-energy lower bounds, and nonstabilizerness-witness construction—can be transferred from a canonical representative to the original Pauli-generator instance.

Background

PauLie transforms the anticommutation graph of a Pauli-generator set into a perfect canonical graph while preserving the isomorphism type of the generated dynamical Lie algebra. The paper observes that perfectness enables several downstream applications, including efficient entanglement-detection schemes, connections to shadow tomography, tight uncertainty relations, strong ground-state-energy lower bounds, and efficiently solvable reduced stabilizer polytopes for certain magic-detection settings.

The unresolved issue is whether, and to what extent, these properties and application guarantees for the canonical representative can be transferred back to the original input instance. This matters because the canonical graph preserves the relevant Lie-algebraic classification but may not preserve the concrete structure required by each application.