有限域上点计数近似的量子算法与困难性
Quantum Algorithms and Hardness for Point-Count Approximation over Finite Fields
- Toyota Central R&D Labs., Inc.（丰田中央研究开发研究所）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对有限域上劳伦多项式的点计数近似问题，提出了一种具有加性精度的量子算法，并证明该问题在支撑矩阵自由变化时为#P困难，明确了量子方法有效性与输入参数的权衡关系。
AI中文摘要：
我们研究有限域上劳伦多项式解的数量近似问题。对于劳伦多项式 $f(x)=\sum_{j=1}^{s}a_jx^{u_j}\in \mathbb{F}_q[x_1^{\pm1},\ldots,x_n^{\pm1}]$，设 $U$ 为其增广支撑矩阵，列向量为 $(1,u_j)$，秩为 $\rho$，$N(f):= \\# \{x\in (\mathbb{F}_q^\times)^n \mid f(x)=0\}$ 为其环面点计数。我们的第一个主要结果是一个量子算法，输出 $\widehat{N}(f)$ 满足 $|\widehat{N}(f) - N(f)| \le \varepsilon q^{n+s/2-\rho}$，成功概率为 $1-\delta$。若 $\rho$ 和 $\\|U\\|_\infty$ 有界，则该算法的经典比特复杂度和量子门复杂度均为 $\mathrm{poly}(n, s, \log q, 1/\varepsilon, \log(1/\delta))$，在一般情形下比相对误差近似具有更精细的分辨率。据我们所知，在所考虑的显式有限域输入模型中，尚无先前算法能以多项式于 $\log q$ 的运行时间达到这种加性精度。Van Dam（arXiv:quant-ph/0405081）曾猜想，在存在反映多项式代数性质的预言机的假设下，存在此类算法。与之相反，我们利用有限域上特征和推导的点计数公式，开发了一种替代方法，无需假设此类预言机存在即可高效近似点的数量。作为第二个主要结果，我们证明，当支撑矩阵 $U$ 作为输入自由变化时，相同的近似问题在随机多项式时间图灵归约下为 #P 困难。因此，综合来看，我们的结果阐明了量子方法的有效性如何依赖于输入多项式的精度尺度与支撑参数之间的权衡关系。
英文摘要：
We study the approximation of the number of solutions of Laurent polynomials over finite fields. For a Laurent polynomial \[f(x)=\sum_{j=1}^{s}a_jx^{u_j}\in \mathbb{F}_q[x_1^{\pm1},\ldots,x_n^{\pm1}], \] let $U$ be its augmented support matrix whose columns are $(1,u_j)$ with rank $ρ$ and $N(f) := \# \{x\in (\mathbb{F}_q^\times)^n \mid f(x)=0\}$ be its torus point count. Our first main result is a quantum algorithm that outputs $\widehat{N}(f)$ satisfying \[ |\widehat{N}(f) - N(f)| \le \varepsilon q^{n+s/2-ρ} \] with success probability $1-δ$. Provided that $ρ$ and $\|U\|_\infty$ are bounded, the algorithm runs in both classical bit and quantum gate complexity $\mathrm{poly}(n, s, \log q, 1/\varepsilon, \log(1/δ))$. It provides finer resolution than relative-error approximations in general settings. To the best of our knowledge, in the explicit finite-field input model considered here, no previous algorithm achieves this additive accuracy with running time polynomial in $\log q$. Van Dam (arXiv:quant-ph/0405081) conjectured the existence of such an algorithm under the assumption of an oracle reflecting the algebraic properties of the polynomial. In contrast, by exploiting a point-counting formula derived from character sums over finite fields, we develop an alternative approach that efficiently approximates the number of points without assuming the existence of such an oracle. As a second main result, we prove that the same approximation problem becomes $\#$P-hard under randomized polynomial-time Turing reductions when the support matrix $U$ varies freely as part of the input. Thus, taken together, our results clarify how the effectiveness of the quantum approach depends on the tradeoff between the accuracy scale and the support parameters of the input polynomial.
