Speaker
Description
Data re-uploading circuits are usually analyzed for scalar data encoded through commuting generators, where the model output is a truncated Fourier series. We study the non-Abelian counterpart in which the uploaded datum is an unknown unitary matrix U ∈ U(N), queried repeatedly between trainable unitary layers and read out through a Born probability.
We formulate the correct center-invariant target space, introduce a balanced degree filtration for polynomial functions on the unitary domain, and prove three obstructions. First, every positive-depth output has genuine balanced degree equal to the number of uploads. Second, at fixed depth the reachable coefficient image is semi-algebraic and nowhere dense in the ambient finite-dimensional coefficient space. Third, even unbounded depth is not uniformly universal: a topological range-pinning obstruction separates the full raw model from valid [0,1]-valued center-invariant targets.
As a showcase of the semi-algebraic image-geometry tools developed here, we also obtain a rigorous non-universality statement for Boson Sampling Born Machines. Overall, the work provides a new geometric framework for studying reachable model images in PQC expressivity theory.
| Topic | Theoretical foundations of quantum learning |
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