Exact complexity of phantomness recognition

Determine which complexity class the phantomness recognition problem belongs to, or identify a specific decision problem to which it is polynomial-time equivalent, thereby settling the computational classification not resolved by the GI-hardness result.

Background

The paper proves that recognizing whether a binary stabilizer code is phantom is GI-hard, including for CSS codes encoding only two logical qubits. GI-hardness establishes a lower bound but does not determine the problem’s precise complexity classification or show polynomial-time equivalence to another standard decision problem.

The unresolved issue is therefore to place phantomness recognition in an appropriate complexity class or to establish a polynomial-time characterization through an equivalence reduction.