半群乘积问题的量子查询复杂度
The quantum query complexity of the semigroup product problem
- University of Technology Sydney（悉尼科技大学）
- Quantinuum, Singapore（Quantinuum新加坡）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究半群乘积问题的量子查询复杂度，引入乘积广度刻画复杂度，对非周期幺半群给出紧界，并改进了一般非周期半群的上界。
AI中文摘要：
我们研究了计算半群乘积 $x_1\cdots x_n$ 的量子查询复杂度，其中一次查询揭示一个输入元素，且乘法表已知。对于大小为 $N-1$ 的有限非周期半群，Aaronson、Grier 和 Schaeffer 的论证给出了 $\sqrt n\\,(N\log(nN+2))^{O(N)}$ 次查询的上界。为了获得反映代数结构的查询界，我们研究了乘积广度 $\beta$：即最小的界，使得每个输入词都有一个至多 $\beta$ 个字母的子序列具有相同的乘积。对于非平凡的有限交换非周期幺半群，有界误差量子查询复杂度为 $\Theta(\min\{n,\sqrt{n\beta}\})$，因此由乘积广度刻画。我们进一步证明，如果这样的幺半群 $M$ 具有非周期指数 $k$（满足对每个 $x\in M$ 有 $x^k=x^{k+1}$ 的最小正整数），则 $\beta=O(k\log(|M|+1)\log\log(|M|+2))$。对于具有稳定偏序且恒等元为最小元素的幺半群，我们证明有界误差量子查询复杂度至多为 $\sqrt{n+1}((\beta+2)\log(n+2))^{O(\log(\beta+2))}$。对于阶为 $N-1$ 的任意有限非周期半群，我们改进了 Aaronson、Grier 和 Schaeffer 的界，得到有界误差量子查询复杂度至多为 \\[ \min\left\{n,\sqrt n\\, \log^{O((N\log(N+2))^{1/3})}(n+2)\right\}. \\] 对半群大小的依赖几乎是紧的：Ambainis 等人的有界深度 Dyck 下界表明，在相关参数范围内，非周期幺半群需要 $\sqrt n\\,2^{\Omega(N^{1/3})}$ 次量子查询。
英文摘要：
We study the quantum query complexity of computing a semigroup product $x_1\cdots x_n$, when one query reveals one input element and the multiplication table is given. For a finite aperiodic semigroup of size $N-1$, the argument of Aaronson, Grier, and Schaeffer gives an upper bound of $\sqrt n\,(N\log(nN+2))^{O(N)}$ queries. To obtain query bounds that reflect algebraic structure, we study the product breadth $β$: the smallest bound such that every input word has a subsequence of at most $β$ letters with the same product. For nontrivial finite commutative aperiodic monoids, the bounded-error quantum query complexity is $Θ(\min\{n,\sqrt{nβ}\})$, and is thus characterized by product breadth. We further show that if such a monoid $M$ has aperiodicity index $k$ (the least positive integer satisfying $x^k=x^{k+1}$ for every $x\in M$), then $β=O(k\log(|M|+1)\log\log(|M|+2))$. For monoids with a stable partial order in which the identity is the minimum element, we prove that the bounded-error quantum query complexity is at most $\sqrt{n+1}((β+2)\log(n+2))^{O(\log(β+2))}$. For arbitrary finite aperiodic semigroups of order $N-1$, we improve the bound of Aaronson, Grier, and Schaeffer, obtaining a bounded-error quantum query complexity of at most \[ \min\left\{n,\sqrt n\, \log^{O((N\log(N+2))^{1/3})}(n+2)\right\}. \] The dependence on semigroup size is nearly tight: the bounded-depth Dyck lower bound of Ambainis et al. yields aperiodic monoids requiring $\sqrt n\,2^{Ω(N^{1/3})}$ quantum queries in the relevant parameter range.
