Extend i-th order tensor characterization to non-diagonal Clifford hierarchy operators

Establish whether non-diagonal operators in the i-th level of the Clifford hierarchy admit an i-th order tensor representation within the quadratic/higher-order tensor framework introduced in this paper (i.e., specified by an i-th order function q and a (i−1)th-order embedding ε), analogous to the diagonal case characterized in Proposition 6.1, and determine the precise conditions under which such a correspondence holds or fails.

Background

The paper proves that diagonal operators in the i-th level of the Clifford hierarchy are precisely described by i-th order tensors—constructed from i-th order functions and (i−1)th-order embeddings—thereby generalizing known results beyond prime-dimensional qudits and connecting the hierarchy to higher-order polynomial maps between groups.

However, the authors explicitly note uncertainty about whether this correspondence extends beyond diagonal operators. Resolving this would clarify the scope of the higher-order tensor formalism in capturing the full Clifford hierarchy, including non-diagonal elements, and strengthen the proposed unification across discrete and continuous-variable models.