Extension of hardness to recognizing symmetry-protected topological phases

Establish whether the quantum computational hardness lower bounds for recognizing topological order proven via shallow pseudorandom unitary constructions extend to recognizing symmetry-protected topological phases of matter under a specified protecting symmetry group.

Background

The paper proves quantum hardness for recognizing topological order under polylogarithmic-depth constraints using pseudorandom unitaries. The authors ask whether similar hardness extends when the phase is protected by a symmetry group, as in symmetry-protected topological (SPT) phases, which are central in condensed matter and quantum information.