Quantum Černý complexity of binary words

Abstract

We introduce the quantum Černý complexity $\mathrm{qc}(w)$ of a binary word $w$: the least dimension $d$ for which there exist quantum channels $A_0,A_1$ on $d\times d$ density matrices and a start state $ρ_0$ such that $w$ is the unique shortest word whose associated channel is constant on the reachable set. We show that $2\le\mathrm{qc}(w)\le\lceil\sqrt{|w|+1}\,\rceil$ for every nonempty $w$, a quadratic saving over the classical analogue, and that constant words are extremal: $\mathrm{qc}(0^m)=\lceil\sqrt{m+1}\,\rceil$. In contrast, $\mathrm{qc}(01^n0)=2$ for every $n\ge 1$, realized by a single qubit whose rotation angle acts as a counter; consequently there is no quantum analogue of the Černý function, and $\mathrm{qc}$ is strongly anti-correlated with intuitive notions of descriptive complexity. We further study the variant $\mathrm{qcp}$ in which the synchronization target is required to be a pure state. We prove that in dimension $2$ no word of length at least $2$ can be a unique shortest synchronizing word with pure target, and we exhibit an explicit qutrit instance, combining a coherent rotation with a measure-and-funnel channel, achieving $\mathrm{qcp}(01^n0)=3$ with target a computational basis state and with synchronization holding universally over all input states. Thus purity of the reset state costs exactly one dimension on this family. We also observe that $\mathrm{qc}$ is computable, by reduction to the first-order theory of the reals.