Quantum Černý complexity of binary words

2026-09-30

AI Summary

This study defines and investigates the quantum Černý complexity of binary words to address the dimensionality requirements of classical synchronizing words under quantum channels. By introducing quantum channels and initial states, combined with density matrix operations and real first-order logic reduction techniques, it establishes the minimal synchronizing dimension and analyzes pure-state target variants. The main contributions include proving that this complexity is bounded between 2 and √(n+1), revealing a quadratic dimensional advantage of quantum over classical settings. Furthermore, the work demonstrates the absence of a quantum analogue to the classical Černý function and quantifies the impact of purity on dimensionality. Finally, it provides optimal constructions for specific word families and confirms the computability of the proposed metric.

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.

Problem

• Quantum Černý complexity
• Synchronizing word
• Quantum channels
• Pure state target
• Binary words

Innovation

• Quantum Černý complexity
• quantum channels
• synchronizing word
• pure state target
• first-order theory of the reals