Algorithmic complexity of computing qc

Determine whether qc(w) is computable in time polynomial in the length of w, and determine whether the decision set {(w,k): qc(w)≤k} belongs to NP or to the existential theory of the reals.

Background

The paper proves that qc is computable by expressing the existence of a suitable quantum-channel instance in the first-order theory of the reals and applying Tarski’s decision procedure. This establishes decidability but does not provide an efficient algorithm.

The unresolved questions concern the computational complexity of evaluating qc and of deciding whether a given word admits a realization in dimension at most k.

References

Remark~\ref{rem:computable} gives computability via real quantifier elimination, which is far from efficient. Is $qc(w)$ computable in time polynomial in $|w|$? Is the set ${(w,k):qc(w)\le k}$ in NP, or $\existsR$?

— Quantum Černý complexity of binary words  (2609.40154 - Lee et al., 30 Sep 2026) in Section Discussion and open problems, subsection “Open problems,” item 2 (Complexity of computing qc)