Partial phantomness for task-specific logical operations

Determine whether either distance obstruction can be avoided when full phantomness is replaced by a task-specific subgroup of logical operations, specifically for SWAP phantomness realizing every permutation of a fixed logical basis or for GHZ-preparation phantomness realizing the transformation $|+\rangle|0\rangle^{\otimes(k-1)}\mapsto (|0^k\rangle+|1^k\rangle)/\sqrt2$.

Background

The no-go theorem concerns full phantomness, namely the realization of the entire standard logical $GL_k(\mathbb F_2)$ action by physical coordinate permutations. The paper asks whether the obstruction persists if only a restricted, task-specific logical subgroup or transformation must be implemented.

The two proposed targets are SWAP phantomness, already studied in the paper’s GI-hardness result for logical relabelling, and GHZ-preparation phantomness, requiring only the stated logical state transformation.