Showcasing a Barren Plateau Theory Beyond the Dynamical Lie Algebra

TL;DR

This paper extends the understanding of barren plateaus in variational quantum algorithms by deriving an exact variance formula for arbitrary states and measurements in matchgate circuits, revealing new phenomena beyond the Lie algebra constraint.

Contribution

It provides a generalized variance formula for matchgate circuits beyond the Lie algebra assumption, introducing the concept of Lie group modules and their role in barren plateaus.

Findings

Variance does not necessarily vanish inversely with Lie algebra dimension.

Generalized global operators lead to barren plateaus.

Connection between generalized entanglement and fermionic entanglement measures.

Abstract

Barren plateaus have emerged as a pivotal challenge for variational quantum computing. Our understanding of this phenomenon underwent a transformative shift with the recent introduction of a Lie algebraic theory capable of explaining most sources of barren plateaus. However, this theory requires either initial states or observables that lie in the circuit's Lie algebra. Focusing on parametrized matchgate circuits, in this work we are able to go beyond this assumption and provide an exact formula for the loss function variance that is valid for arbitrary input states and measurements. Our results reveal that new phenomena emerge when the Lie algebra constraint is relaxed. For instance, we find that the variance does not necessarily vanish inversely with the Lie algebra's dimension. Instead, this measure of expressiveness is replaced by a generalized expressiveness quantity: The dimension of the Lie group modules. By characterizing the operators in these modules as products of Majorana operators, we can introduce a precise notion of generalized globality and show that measuring generalized-global operators leads to barren plateaus. Our work also provides operational meaning to the generalized entanglement as we connect it with known fermionic entanglement measures, and show that it satisfies a monogamy relation. Finally, while parameterized matchgate circuits are not efficiently simulable in general, our results suggest that the structure allowing for trainability may also lead to classical simulability.