Dimension-two characterization and the Thue–Morse prefix

Determine whether qc(01101001)=2 and characterize the binary words w for which the mixed-target quantum Černý complexity satisfies qc(w)=2, including whether every word with at least one alternation and no long constant power belongs to this class.

Background

The paper studies qc(w), the least quantum-system dimension in which w is the unique shortest synchronizing word, and proves that qc(w)=2 for every word of the form 01n0. It also proves that the pure-target variant has dimension two only for the single-letter words 0 and 1.

For the eight-bit Thue–Morse prefix 01101001, the general upper bound gives qc(01101001)≤3, but the authors conjecture that dimension two suffices. They further ask for a characterization of all words realizable in dimension two and specifically identify as unresolved whether all words having at least one alternation and no long constant power have qc equal to two.

References

We conjecture $qc(01101001)=2$; the bound $qc(01101001)\le3$ follows from Theorem~\ref{thm:kmp}. More generally, characterize ${w:qc(w)=2}$. By Theorem~\ref{thm:d2pure} the analogous pure-target class is trivial, ${0,1}$, so the interest is genuinely in the mixed-target case. A dimension count in the Bloch picture (both letters must act by singular Bloch maps when $w$ begins and ends with distinct letters, and collapse is governed by kernel--image incidences) suggests the class is large, but we do not know whether, e.g., all words with at least one alternation and no long constant power belong to it.

— Quantum Černý complexity of binary words  (2609.40154 - Lee et al., 30 Sep 2026) in Section Discussion and open problems, subsection “Open problems,” item 1 (Thue–Morse prefix)

What is $qc(w)$ for a uniformly random $w$ of length $m$? The classical counting argument gives $rc(w)\gtrsim m/(2\log m)$ for most $w$, but it is consistent with our results that $qc(w)=2$ for almost all $w$.

— Quantum Černý complexity of binary words  (2609.40154 - Lee et al., 30 Sep 2026) in Section Discussion and open problems, subsection “Open problems,” item 3 (Typical words)