二元词的量子Černý复杂度
Quantum Černý complexity of binary words
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中文总结 AI 辅助
本文引入二元词的量子Černý复杂度，证明其界为二次方节省，并揭示与经典情形及描述复杂性的反直觉关系，同时研究纯态目标变体及可计算性。
中文摘要 AI 辅助
我们引入了二元词$w$的量子Černý复杂度$\mathrm{qc}(w)$：即存在量子信道$A_0,A_1$作用于$d\times d$密度矩阵以及初始状态$\rho_0$，使得$w$是唯一最短的、其关联信道在可达集上为常数的词的最小维度$d$。我们证明对于每个非空词$w$，有$2\le\mathrm{qc}(w)\le\lceil\sqrt{|w|+1}\\,\rceil$，这相对于经典类比是二次方的节省，并且常数词是极端的：$\mathrm{qc}(0^m)=\lceil\sqrt{m+1}\\,\rceil$。相反，对于每个$n\ge 1$，$\mathrm{qc}(01^n0)=2$，由单个量子比特实现，其旋转角度充当计数器；因此不存在Černý函数的量子类比，且$\mathrm{qc}$与描述复杂性的直观概念强烈负相关。我们进一步研究了变体$\mathrm{qcp}$，其中同步目标要求为纯态。我们证明在维度2中，长度至少为2的词不能是唯一最短的具有纯目标的同步词，并且我们展示了一个显式的qutrit实例，结合相干旋转与测量-漏斗信道，实现了$\mathrm{qcp}(01^n0)=3$，目标为计算基态，且同步对所有输入状态普遍成立。因此，重置态的纯度在该族上恰好花费一个维度的代价。我们还观察到$\mathrm{qc}$是可计算的，通过归约到实数的一阶理论。
英文摘要
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.
