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Barren plateaus have emerged as a pivotal challenge for variational quantumcomputing. Our understanding of this phenomenon underwent a transformativeshift with the recent introduction of a Lie algebraic theory capable ofexplaining most sources of barren plateaus. However, this theory requireseither initial states or observables that lie in the circuit's Lie algebra.Focusing on parametrized matchgate circuits, in this work we are able to gobeyond this assumption and provide an exact formula for the loss functionvariance that is valid for arbitrary input states and measurements. Our resultsreveal that new phenomena emerge when the Lie algebra constraint is relaxed.For instance, we find that the variance does not necessarily vanish inverselywith the Lie algebra's dimension. Instead, this measure of expressiveness isreplaced by a generalized expressiveness quantity: The dimension of the Liegroup modules. By characterizing the operators in these modules as products ofMajorana operators, we can introduce a precise notion of generalized globalityand show that measuring generalized-global operators leads to barren plateaus.Our work also provides operational meaning to the generalized entanglement aswe connect it with known fermionic entanglement measures, and show that itsatisfies a monogamy relation. Finally, while parameterized matchgate circuitsare not efficiently simulable in general, our results suggest that thestructure allowing for trainability may also lead to classical simulability.

Showcasing a Barren Plateau Theory Beyond the Dynamical Lie Algebra