Quadratic upper bound for the pure-target variant

Establish whether every binary word w satisfies the pure-target bound qc(w)=O(sqrt(|w|)), thereby obtaining a pure-target analogue of the general quantum Černý complexity upper bound.

Background

The paper constructs qutrit realizations with pure targets for the family 01n0, demonstrating a substantial improvement over the classical bound but not a general square-root bound.

The general upper-bound construction for qc uses the maximally mixed state as its synchronization target and does not extend directly to the pure-target setting. The open question is whether a comparable quadratic dimensional saving holds universally for pure targets.

References

Does a general quadratic-type saving $qc(w)\le O(\sqrt{|w|})$ hold, i.e. a pure-target analogue of Theorem~\ref{thm:kmp}? Our proof of Theorem~\ref{thm:kmp} places the target at the maximally mixed state and does not adapt.

— Quantum Černý complexity of binary words  (2609.40154 - Lee et al., 30 Sep 2026) in Section Discussion and open problems, subsection “Open problems,” item 5 (Pure-target savings)