Characterize the composition of block-wise polar factors into full model layers

Determine how block-wise polar factors compiled into exchange-only networks can be composed to implement a complete linear layer, including the associated input encoding, normalization, and placement of the positive factors.

Background

The appendix considers decomposing a real weight block into an orthogonal polar factor and a positive factor. The exchange-only network is used to compile the orthogonal factor, while the positive factor is retained as a classical operation.

The numerical study addresses only individual small blocks. It does not resolve how such block-wise decompositions should be assembled into a full neural-network layer or how the resulting signed linear map should be represented and normalized on a quantum register.