Remove the square-root dimension factor from the subnormalization

Determine whether the subnormalization factor alpha=2sqrt{N} can be reduced by removing the sqrt{N} factor within a generator-based, factorization-free block-encoding construction for generator-representable semiseparable matrices.

Background

The proposed circuit block-encodes a generator-representable semiseparable matrix with subnormalization factor alpha=2sqrt{N}. The authors note that this factor is not generally optimal relative to the spectral norm: for unit-norm generators, the matrix norm can be bounded independently of N, so the encoding overhead may grow as Theta(sqrt{N}).

The sqrt{N} factor arises from the final Hadamard layer that superposes N replicated column copies. The unresolved question is whether a different construction that still uses the generator representation and avoids matrix factorization can eliminate this normalization loss.