Dimension classification for small quantum circuits

Establish whether an exact dimension classification analogous to the classical SIZE(α·2^n/n) result holds for quantum circuit classes, for example by determining the PSPACE-dimension or GapP-dimension (or Hausdorff dimension) of BQSIZE(α·2^n/n) for α in [0,1].

Background

Classically, SIZE(α·2^n/n) has exact dimension α (in several resource-bounded senses). The paper proves a zero strong-dimension result for sublinear-in-2^n quantum circuit classes using GapP-martingales but does not provide an exact classification at fixed α.

A quantum analogue would refine current dimension-zero statements into a precise scale, potentially depending on the chosen universal gate set.